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The task is to determine the transitivity of a graph.
Transitivity is the ratio of the number of triangles in the graph to the number of connected triples of nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, and 25. The graph contains edges: (1, 22), (1, 4), (5, 22), (4, 5), (2, 25), (24, 25), (3, 24), (3, 6), (19, 24), (5, 23), (5, 9), (7, 23), (7, 9), (7, 8), (10, 20), (10, 11), (11, 13), (12, 13), (13, 14), (14, 15), (16, 17), (17, 18), (19, 21).
Question: What is the transitivity of the graph?
You need to format your answer as a float number. | 0 | graph_transitivity |
The task is to determine if the graph is bipartite.
A bipartite graph is a graph whose nodes can be divided into two disjoint sets such that no two graph vertices within the same set are adjacent.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, and 5. The graph contains edges: (1, 3), (1, 4), (2, 4), (2, 5).
Question: Is the graph bipartite?
Your answer should be Yes or No. | Yes | graph_is_bipartite |
The task is to determine if the graph is bipartite.
A bipartite graph is a graph whose nodes can be divided into two disjoint sets such that no two graph vertices within the same set are adjacent.
Here is a directed graph. In the graph, (u, v) means that there is an edge from node u to node v. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, and 25. The graph contains edges: (1, 4), (1, 12), (1, 3), (1, 10), (1, 8), (1, 18), (1, 23), (1, 5), (1, 19), (1, 9), (1, 25), (1, 17), (1, 7), (1, 14), (1, 16), (1, 24), (4, 12), (4, 3), (4, 8), (4, 13), (4, 15), (4, 23), (4, 19), (4, 9), (4, 11), (4, 25), (4, 21), (4, 17), (4, 7), (4, 14), (4, 16), (4, 1), (4, 24), (12, 4), (12, 3), (12, 10), (12, 8), (12, 13), (12, 2), (12, 15), (12, 23), (12, 5), (12, 19), (12, 9), (12, 11), (12, 25), (12, 21), (12, 17), (12, 7), (12, 14), (12, 16), (12, 1), (12, 24), (3, 4), (3, 12), (3, 10), (3, 8), (3, 15), (3, 18), (3, 23), (3, 5), (3, 19), (3, 9), (3, 22), (3, 11), (3, 25), (3, 21), (3, 17), (3, 7), (3, 14), (3, 16), (3, 1), (3, 24), (10, 4), (10, 12), (10, 3), (10, 6), (10, 8), (10, 13), (10, 2), (10, 15), (10, 18), (10, 23), (10, 19), (10, 9), (10, 25), (10, 21), (10, 17), (10, 7), (10, 14), (10, 16), (10, 1), (10, 24), (8, 12), (8, 3), (8, 10), (8, 6), (8, 13), (8, 2), (8, 23), (8, 19), (8, 17), (8, 7), (8, 14), (8, 16), (8, 1), (18, 12), (18, 3), (18, 10), (18, 8), (18, 15), (18, 23), (18, 9), (18, 14), (18, 16), (23, 4), (23, 12), (23, 3), (23, 10), (23, 6), (23, 8), (23, 13), (23, 2), (23, 15), (23, 18), (23, 19), (23, 25), (23, 21), (23, 17), (23, 7), (23, 14), (23, 16), (23, 1), (5, 3), (5, 9), (5, 11), (5, 16), (5, 1), (5, 24), (19, 4), (19, 12), (19, 3), (19, 10), (19, 8), (19, 2), (19, 18), (19, 23), (19, 9), (19, 25), (19, 21), (19, 17), (19, 7), (19, 14), (19, 16), (19, 1), (19, 24), (9, 4), (9, 12), (9, 3), (9, 10), (9, 8), (9, 2), (9, 18), (9, 19), (9, 22), (9, 11), (9, 25), (9, 17), (9, 7), (9, 14), (9, 16), (9, 1), (9, 24), (25, 4), (25, 12), (25, 3), (25, 10), (25, 6), (25, 8), (25, 13), (25, 15), (25, 18), (25, 23), (25, 19), (25, 9), (25, 21), (25, 17), (25, 7), (25, 14), (25, 16), (25, 1), (17, 4), (17, 19), (17, 9), (17, 7), (17, 16), (17, 1), (7, 4), (7, 12), (7, 3), (7, 10), (7, 8), (7, 13), (7, 2), (7, 18), (7, 23), (7, 19), (7, 9), (7, 25), (7, 21), (7, 17), (7, 20), (7, 14), (7, 16), (7, 1), (7, 24), (14, 4), (14, 12), (14, 3), (14, 10), (14, 6), (14, 8), (14, 15), (14, 18), (14, 23), (14, 19), (14, 9), (14, 25), (14, 7), (14, 16), (14, 1), (14, 24), (16, 4), (16, 12), (16, 3), (16, 10), (16, 6), (16, 8), (16, 18), (16, 23), (16, 5), (16, 19), (16, 9), (16, 11), (16, 25), (16, 17), (16, 7), (16, 14), (16, 1), (16, 24), (24, 4), (24, 12), (24, 3), (24, 10), (24, 8), (24, 19), (24, 9), (24, 22), (24, 11), (24, 17), (24, 14), (24, 16), (24, 1), (2, 10), (2, 8), (2, 13), (2, 25), (2, 7), (2, 14), (2, 1), (13, 4), (13, 12), (13, 10), (13, 8), (13, 2), (13, 15), (13, 19), (13, 25), (13, 7), (13, 14), (13, 1), (15, 4), (15, 12), (15, 3), (15, 10), (15, 6), (15, 18), (15, 23), (15, 25), (15, 21), (15, 17), (15, 14), (22, 3), (22, 19), (22, 9), (22, 24), (11, 4), (11, 3), (11, 5), (11, 19), (11, 25), (11, 7), (11, 16), (11, 1), (11, 24), (21, 12), (21, 3), (21, 10), (21, 6), (21, 8), (21, 15), (21, 23), (21, 19), (21, 25), (21, 17), (21, 7), (21, 1), (6, 4), (6, 12), (6, 10), (6, 8), (6, 17), (6, 20), (6, 7), (6, 24), (20, 4), (20, 7).
Question: Is the graph bipartite?
Your answer should be Yes or No. | No | graph_is_bipartite |
The task is to determine the degree assortativity coefficient of a graph.
The degree assortativity coefficient is a measure of the correlation between the degrees of connected nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, and 5. The graph contains edges: (1, 5), (1, 4), (3, 5), (2, 3).
Question: What is the degree assortativity of the graph?
You need to format your answer as a float number. | -0.3333333333333333 | graph_degree_assortativity |
The task is to determine the number of weakly connected components in a directed graph.
A weakly connected component is a maximal subgraph where every node is reachable from every other node when ignoring the direction of edges.
Here is a directed graph. In the graph, (u, v) means that there is an edge from node u to node v. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, and 22. The graph contains edges: (1, 2), (1, 22), (2, 12), (2, 21), (2, 4), (2, 1), (22, 21), (22, 19), (22, 1), (12, 9), (12, 2), (21, 17), (21, 18), (21, 2), (21, 22), (4, 6), (4, 17), (4, 18), (4, 2), (3, 5), (5, 3), (5, 17), (5, 18), (6, 15), (17, 5), (17, 21), (17, 4), (17, 19), (18, 5), (18, 21), (18, 4), (18, 19), (15, 9), (15, 6), (7, 11), (7, 8), (11, 7), (8, 7), (9, 13), (9, 14), (9, 15), (13, 16), (13, 20), (13, 9), (14, 16), (14, 20), (14, 9), (10, 16), (10, 20), (16, 10), (16, 13), (16, 14), (20, 10), (20, 13), (20, 14), (19, 17), (19, 18), (19, 22).
Question: How many weakly connected components are there in the graph?
| 2 | graph_weakly_connected_number |
The task is to determine if the graph is Eulerian.
An Eulerian graph is a graph that contains an Eulerian circuit, which is a cycle that visits every edge exactly once.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, and 5. The graph contains edges: (1, 5), (1, 2), (2, 5), (3, 5), (4, 5), (3, 4).
Question: Is the graph Eulerian?
Your answer should be Yes or No. | Yes | graph_is_eularian |
The task is to determine the minimum edge covering of a graph.
An edge cover is a set of edges such that every vertex in the graph is incident to at least one edge in the set. The minimum edge cover is the edge cover with the smallest number of edges.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, and 28. The graph contains edges: (1, 21), (1, 6), (6, 23), (2, 18), (2, 28), (7, 18), (22, 28), (3, 23), (11, 23), (23, 25), (4, 27), (7, 27), (5, 9), (5, 14), (13, 14), (7, 17), (7, 24), (8, 16), (12, 16), (15, 16), (16, 26), (10, 12), (10, 19), (19, 22), (13, 20).
Question: What is the minimum edge covering of the graph?
You need to format your answer as a list of edges in ascending dictionary order, e.g., [(u1, v1), (u2, v2), ..., (un, vn)]. | [(1,21),(2,18),(5,9),(6,1),(7,17),(12,10),(14,5),(16,8),(16,15),(20,13),(22,19),(23,3),(23,11),(24,7),(25,23),(26,16),(27,4),(28,2)] | graph_min_edge_covering |
The task is to determine the effective size of a node in a graph.
The effective size of a node is the number of nodes that are not directly connected to it, but are connected to its neighbors.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. The graph contains edges: (1, 4), (1, 3), (1, 6), (1, 5), (1, 7), (3, 4), (4, 6), (4, 5), (3, 6), (3, 8), (6, 8), (5, 7), (5, 9), (5, 10), (7, 9), (7, 10), (2, 8), (9, 10).
Question: What is the effective size of node 1 in the graph?
You need to format your answer as a float number. | 3.0 | graph_effective_size |
The task is to determine the resource allocation index of two nodes in a graph.
The resource allocation index of two nodes is the sum of the inverse of the degrees of the common neighbors of the two nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, and 33. The graph contains edges: (1, 4), (1, 14), (1, 16), (1, 15), (4, 14), (4, 16), (4, 15), (4, 27), (14, 16), (14, 31), (14, 15), (14, 27), (15, 16), (16, 27), (15, 27), (2, 3), (2, 32), (2, 33), (3, 28), (3, 32), (3, 33), (32, 33), (17, 27), (22, 27), (27, 31), (5, 7), (6, 7), (8, 11), (8, 12), (8, 13), (8, 9), (8, 10), (8, 30), (9, 11), (10, 11), (11, 12), (11, 13), (9, 12), (10, 12), (12, 13), (9, 13), (10, 13), (9, 10), (26, 30), (17, 31), (17, 22), (17, 18), (19, 24), (19, 29), (19, 20), (19, 25), (24, 25), (20, 29), (21, 29), (25, 29), (20, 25), (20, 21), (23, 26).
Question: What is the resource allocation index between node 5 and node 29?
You need to format your answer as a float number. | 0 | graph_resource_allocation_index |
The task is to determine the breadth-first search (BFS) traversal order given a starting node.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, and 23. The graph contains edges: (1, 17), (3, 17), (4, 17), (12, 17), (13, 17), (14, 17), (17, 20), (17, 18), (2, 15), (2, 18), (2, 4), (2, 19), (2, 13), (2, 14), (2, 5), (4, 15), (11, 15), (14, 15), (15, 20), (15, 18), (6, 18), (7, 18), (8, 18), (12, 18), (13, 18), (18, 20), (18, 19), (3, 4), (4, 10), (4, 20), (4, 21), (4, 8), (4, 12), (4, 13), (4, 9), (4, 5), (4, 7), (3, 19), (6, 19), (19, 22), (8, 13), (12, 13), (13, 20), (13, 14), (3, 14), (7, 14), (8, 14), (9, 14), (10, 14), (12, 14), (5, 23), (5, 20), (3, 20), (3, 8), (8, 20), (9, 20), (10, 20), (12, 20), (16, 20), (20, 22), (7, 8), (8, 12), (8, 9), (10, 23), (10, 11), (7, 21), (9, 21), (16, 21), (7, 12).
Question: What is the breadth-first search (BFS) traversal order for the starting node 5?
You need to format your answer as a list of edges in ascending dictionary order, e.g., [(u1, v1), (u2, v2), ..., (un, vn)]. | [(5,2),(5,4),(5,20),(5,23),(2,13),(2,14),(2,15),(2,18),(2,19),(4,3),(4,7),(4,8),(4,9),(4,10),(4,12),(4,17),(4,21),(20,16),(20,22),(15,11),(18,6),(17,1)] | graph_bfs |
The task is to find the number of triangles that include a specific node as one vertex.
A triangle is a set of three nodes that are all connected to each other.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and 21. The graph contains edges: (1, 2), (1, 3), (1, 5), (1, 6), (1, 7), (1, 8), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (3, 4), (3, 7), (3, 9), (3, 10), (5, 6), (5, 11), (6, 7), (6, 11), (6, 12), (4, 7), (4, 9), (4, 10), (9, 10), (9, 13), (9, 14), (10, 13), (10, 14), (10, 15), (11, 18), (11, 12), (13, 14), (13, 16), (13, 17), (14, 16), (15, 16), (18, 19), (18, 20), (18, 21), (16, 17), (19, 20), (19, 21).
Question: How many triangles include node 6 in the graph?
| 7 | graph_triangles |
The task is to determine the constraint of a node in a graph.
The constraint of a node is a measure of how much the node is constrained by its neighbors.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, and 23. The graph contains edges: (1, 10), (1, 6), (1, 8), (1, 18), (5, 10), (9, 10), (10, 22), (2, 6), (6, 21), (6, 8), (6, 16), (3, 8), (5, 8), (8, 14), (8, 9), (8, 18), (8, 23), (2, 18), (3, 18), (5, 18), (7, 18), (9, 18), (11, 18), (12, 18), (13, 18), (14, 18), (15, 18), (16, 18), (17, 18), (18, 21), (18, 22), (2, 21), (2, 16), (7, 21), (15, 21), (16, 21), (21, 22), (21, 23), (5, 16), (16, 22), (3, 14), (3, 20), (3, 22), (3, 23), (9, 14), (4, 20), (11, 20), (20, 22), (11, 22), (12, 22), (5, 23), (9, 23), (12, 23), (15, 23), (4, 7), (5, 9), (9, 19), (11, 19), (12, 13), (13, 17).
Question: What is the constraint of node 5 in the graph?
You need to format your answer as a float number. | 0.28932109591521793 | graph_constraint |
The task is to determine the minimum spanning tree of a graph.
A minimum spanning tree is a subset of the edges that connects all vertices in the graph with the minimum possible total edge weight. If not specified, all edges have equal edge weights
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. The graph contains edges: (1, 2), (1, 7), (2, 8), (7, 11), (3, 6), (4, 6), (5, 6), (6, 12), (4, 10), (9, 11).
Question: What is the minimum spanning tree of the graph?
You need to format your answer as a list of edges in ascending dictionary order, e.g., [(u1, v1), (u2, v2), ..., (un, vn)]. | [(1,2),(1,7),(2,8),(3,6),(4,6),(4,10),(5,6),(6,12),(7,11),(9,11)] | graph_minimum_spanning_tree |
The task is to find all bridges of a graph.
A bridge is an edge in a graph whose removal increases the number of connected components.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, and 30. The graph contains edges: (1, 16), (1, 21), (15, 16), (2, 29), (2, 5), (3, 29), (5, 24), (4, 6), (6, 19), (17, 19), (7, 23), (7, 9), (8, 23), (20, 23), (9, 11), (8, 30), (10, 12), (10, 26), (12, 13), (12, 14), (25, 26), (26, 27), (17, 28), (17, 18), (20, 22).
Question: What are the bridges of the graph?
You need to format your answer as a list of edges in ascending dictionary order, e.g., [(u1, v1), (u2, v2), ..., (un, vn)]. | [(1,16),(1,21),(2,5),(2,29),(3,29),(4,6),(5,24),(6,19),(7,9),(7,23),(8,23),(8,30),(9,11),(10,12),(10,26),(12,13),(12,14),(15,16),(17,18),(17,19),(17,28),(20,22),(20,23),(25,26),(26,27)] | graph_bridges |
The task is to determine the density of the graph.
Density is defined as the ratio of the number of edges in the graph to the number of possible edges. For undirected graphs, the density is 2*m/(n*(n-1)), where m is the edge number and n is the node number. For directed graphs, the density is m/(n*(n-1)).
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, and 32. The graph contains edges: (1, 23), (2, 23), (23, 24), (2, 24), (24, 25), (3, 29), (3, 4), (3, 20), (4, 29), (29, 30), (4, 20), (4, 31), (20, 31), (30, 31), (5, 16), (5, 18), (5, 17), (14, 16), (16, 18), (16, 17), (17, 18), (18, 19), (13, 17), (14, 17), (17, 21), (6, 26), (9, 26), (10, 26), (26, 27), (7, 8), (7, 9), (7, 11), (8, 9), (8, 10), (8, 11), (9, 10), (9, 11), (9, 27), (9, 28), (10, 11), (10, 27), (10, 28), (27, 28), (12, 13), (12, 21), (13, 21), (14, 15), (22, 30), (22, 32).
Question: What is the density of the graph?
You need to format your answer as a float number. | 0.09879032258064516 | graph_density |
The task is to determine the dominating set of a graph.
A dominating set is a subset of nodes such that every node in the graph is either in the set or adjacent to a node in the set.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, and 34. The graph contains edges: (1, 2), (1, 3), (1, 4), (2, 17), (2, 22), (3, 20), (3, 17), (3, 4), (3, 22), (4, 23), (4, 20), (11, 17), (12, 17), (17, 20), (17, 18), (17, 22), (18, 22), (12, 20), (19, 20), (20, 23), (5, 21), (5, 29), (5, 30), (5, 31), (21, 32), (21, 33), (21, 34), (21, 29), (21, 30), (29, 32), (29, 34), (29, 30), (30, 34), (30, 31), (6, 28), (6, 7), (6, 14), (7, 28), (14, 28), (25, 28), (26, 28), (7, 26), (7, 14), (15, 26), (24, 26), (25, 26), (8, 9), (8, 10), (8, 11), (8, 15), (8, 16), (9, 10), (9, 11), (9, 12), (9, 13), (10, 13), (11, 12), (11, 16), (11, 18), (15, 24), (15, 25), (16, 27), (12, 13), (12, 19), (18, 27), (24, 25), (32, 33), (32, 34), (33, 34).
Question: What is the dominating set of the graph?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [1,5,6,8,12,18,23,24,32] | graph_dominating_set |
The task is to determine the global efficiency of a graph.
Global efficiency is the average of the inverse shortest path lengths between all pairs of nodes in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, and 6. The graph contains edges: (1, 6), (1, 5), (3, 6), (2, 4), (3, 4).
Question: What is the global efficiency of the graph?
You need to format your answer as a float number. | 0.5800000000000001 | graph_global_efficiency |
The task is to determine the barycenter of a graph.
The barycenter of a graph is also called the median. It includes the node that minimizes the sum of distances to all other nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, and 7. The graph contains edges: (1, 5), (1, 6), (1, 2), (1, 4), (2, 5), (5, 6), (2, 6), (2, 4), (3, 4), (3, 7).
Question: What is the barycenter of the graph?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [1,2,4] | graph_barycenter |
The task is to determine common neighbors between two nodes in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, and 30. The graph contains edges: (1, 2), (1, 4), (1, 6), (1, 9), (1, 12), (1, 14), (1, 16), (1, 17), (1, 19), (1, 20), (1, 21), (1, 23), (1, 25), (1, 10), (1, 13), (1, 24), (1, 11), (1, 26), (1, 30), (1, 8), (1, 27), (1, 5), (1, 7), (1, 28), (1, 29), (1, 15), (1, 22), (1, 3), (2, 4), (2, 6), (2, 9), (2, 12), (2, 14), (2, 16), (2, 17), (2, 19), (2, 20), (2, 21), (2, 23), (2, 25), (2, 13), (2, 26), (2, 30), (2, 8), (2, 27), (2, 7), (2, 28), (2, 29), (2, 15), (2, 22), (2, 3), (3, 4), (4, 6), (4, 9), (4, 12), (4, 14), (4, 16), (4, 17), (4, 19), (4, 20), (4, 21), (4, 23), (4, 25), (4, 10), (4, 13), (4, 24), (4, 11), (4, 26), (4, 30), (4, 8), (4, 18), (4, 27), (4, 5), (4, 7), (4, 28), (4, 22), (6, 9), (6, 12), (6, 14), (6, 16), (6, 17), (6, 19), (6, 20), (6, 21), (6, 23), (6, 25), (6, 10), (6, 13), (6, 11), (6, 26), (6, 30), (6, 8), (6, 18), (6, 7), (6, 28), (6, 29), (6, 15), (5, 9), (7, 9), (8, 9), (9, 12), (9, 14), (9, 16), (9, 17), (9, 19), (9, 20), (9, 21), (9, 23), (9, 25), (9, 10), (9, 24), (9, 11), (9, 26), (9, 30), (9, 18), (9, 27), (9, 29), (9, 15), (9, 22), (5, 12), (7, 12), (8, 12), (10, 12), (11, 12), (12, 14), (12, 16), (12, 17), (12, 19), (12, 20), (12, 21), (12, 23), (12, 25), (12, 13), (12, 24), (12, 26), (12, 30), (12, 18), (12, 15), (12, 22), (3, 14), (5, 14), (7, 14), (8, 14), (11, 14), (14, 16), (14, 17), (14, 19), (14, 20), (14, 23), (14, 24), (14, 26), (14, 30), (14, 18), (14, 27), (14, 28), (14, 15), (14, 22), (3, 16), (5, 16), (7, 16), (13, 16), (16, 17), (16, 19), (16, 20), (16, 21), (16, 23), (16, 25), (16, 24), (16, 26), (16, 18), (16, 28), (16, 29), (3, 17), (7, 17), (10, 17), (13, 17), (15, 17), (17, 19), (17, 21), (17, 23), (17, 25), (17, 26), (17, 30), (17, 18), (17, 22), (3, 19), (8, 19), (10, 19), (11, 19), (13, 19), (15, 19), (19, 21), (19, 23), (19, 24), (19, 26), (19, 22), (5, 21), (8, 21), (18, 21), (21, 23), (21, 22), (3, 23), (5, 23), (7, 23), (11, 23), (13, 23), (18, 23), (22, 23), (23, 24), (23, 27), (23, 29), (3, 25), (7, 25), (15, 25), (10, 13), (13, 27), (8, 11), (28, 30), (8, 18).
Question: What are the common neighbors between node 12 and node 17?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [1,2,4,6,7,9,10,13,14,15,16,18,19,21,22,23,25,26,30] | graph_common_neighbor |
The task is to determine the degree assortativity coefficient of a graph.
The degree assortativity coefficient is a measure of the correlation between the degrees of connected nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. The graph contains edges: (1, 2), (1, 3), (2, 3), (2, 4), (3, 4), (5, 6), (5, 8), (5, 9), (5, 10), (6, 7), (8, 9), (8, 10), (7, 9), (9, 11), (9, 10).
Question: What is the degree assortativity of the graph?
You need to format your answer as a float number. | -0.24060150375939737 | graph_degree_assortativity |
The task is to determine the breadth-first search (BFS) traversal order given a starting node.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, and 28. The graph contains edges: (1, 22), (1, 17), (13, 17), (2, 10), (5, 10), (3, 8), (8, 16), (4, 18), (12, 18), (14, 18), (5, 14), (6, 24), (6, 28), (9, 28), (25, 28), (7, 11), (7, 13), (11, 20), (16, 20), (9, 27), (23, 27), (15, 21), (21, 23), (19, 26).
Question: What is the breadth-first search (BFS) traversal order for the starting node 14?
You need to format your answer as a list of edges in ascending dictionary order, e.g., [(u1, v1), (u2, v2), ..., (un, vn)]. | [(14,5),(14,18),(5,10),(18,4),(18,12),(10,2)] | graph_bfs |
The task is to determine the transitivity of a graph.
Transitivity is the ratio of the number of triangles in the graph to the number of connected triples of nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and 21. The graph contains edges: (1, 9), (1, 2), (1, 3), (1, 4), (2, 9), (3, 9), (4, 9), (2, 3), (2, 4), (3, 4), (5, 18), (5, 12), (5, 14), (5, 16), (5, 11), (5, 15), (11, 18), (12, 18), (13, 18), (15, 18), (16, 18), (11, 12), (12, 14), (12, 16), (12, 13), (12, 15), (14, 15), (11, 16), (13, 16), (15, 16), (16, 20), (11, 15), (6, 20), (7, 20), (8, 20), (10, 20), (19, 20), (8, 10), (13, 17), (17, 21).
Question: What is the transitivity of the graph?
You need to format your answer as a float number. | 0.7207792207792207 | graph_transitivity |
The task is to find all bridges of a graph.
A bridge is an edge in a graph whose removal increases the number of connected components.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, and 5. The graph contains edges: (1, 4), (4, 5), (2, 3), (3, 5).
Question: What are the bridges of the graph?
You need to format your answer as a list of edges in ascending dictionary order, e.g., [(u1, v1), (u2, v2), ..., (un, vn)]. | [(1,4),(2,3),(3,5),(4,5)] | graph_bridges |
The task is to find the number of triangles that include a specific node as one vertex.
A triangle is a set of three nodes that are all connected to each other.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, and 19. The graph contains edges: (1, 9), (1, 17), (9, 16), (9, 12), (10, 17), (2, 5), (2, 6), (5, 15), (6, 11), (3, 7), (7, 16), (4, 19), (4, 13), (12, 19), (18, 19), (12, 13), (13, 15), (8, 11), (10, 11), (10, 16), (8, 12), (14, 18).
Question: How many triangles include node 17 in the graph?
| 0 | graph_triangles |
The task is to determine the transitivity of a graph.
Transitivity is the ratio of the number of triangles in the graph to the number of connected triples of nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, and 25. The graph contains edges: (1, 2), (1, 4), (1, 6), (1, 5), (1, 7), (2, 3), (2, 4), (2, 5), (4, 18), (4, 23), (4, 12), (4, 5), (6, 15), (6, 8), (6, 7), (7, 15), (7, 25), (3, 10), (3, 11), (3, 12), (8, 10), (10, 14), (10, 13), (10, 15), (11, 12), (12, 23), (17, 18), (18, 19), (18, 20), (18, 21), (18, 22), (18, 23), (8, 15), (13, 15), (14, 15), (8, 13), (13, 25), (13, 14), (9, 16), (14, 16), (17, 19), (17, 21), (17, 22), (19, 21), (19, 22), (20, 21), (21, 24), (20, 24).
Question: What is the transitivity of the graph?
You need to format your answer as a float number. | 0.484472049689441 | graph_transitivity |
The task is to determine common neighbors between two nodes in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and 21. The graph contains edges: (1, 3), (1, 4), (1, 5), (3, 15), (3, 6), (3, 5), (3, 16), (4, 6), (4, 5), (5, 15), (5, 6), (5, 16), (2, 7), (2, 8), (7, 19), (7, 18), (7, 8), (7, 20), (7, 9), (8, 9), (6, 15), (15, 16), (6, 16), (18, 19), (9, 20), (12, 20), (20, 21), (9, 21), (12, 21), (13, 21), (10, 13), (10, 17), (10, 14), (11, 13), (13, 14), (11, 14).
Question: What are the common neighbors between node 8 and node 9?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [7] | graph_common_neighbor |
The task is to determine the number of connected components in the graph.
A connected component is a subgraph where any two nodes are connected to each other by paths.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, and 7. The graph contains edges: (1, 3), (2, 3), (2, 4), (4, 6), (4, 7), (5, 7).
Question: How many connected components are there in the graph?
| 1 | graph_connected_component_number |
The task is to find the number of triangles that include a specific node as one vertex.
A triangle is a set of three nodes that are all connected to each other.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, and 23. The graph contains edges: (1, 9), (1, 7), (1, 6), (1, 15), (6, 9), (8, 9), (6, 7), (7, 15), (6, 12), (6, 8), (6, 15), (12, 15), (15, 22), (2, 13), (3, 13), (3, 23), (3, 16), (16, 23), (4, 14), (4, 19), (4, 5), (5, 14), (14, 20), (14, 17), (5, 19), (11, 19), (18, 19), (19, 20), (10, 12), (12, 22), (10, 21), (17, 20), (18, 20).
Question: How many triangles include node 5 in the graph?
| 2 | graph_triangles |
The task is to determine the Jaccard coefficient of two nodes in a graph.
The Jaccard coefficient is the size of the intersection divided by the size of the union of the neighbors of the two nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. The graph contains edges: (1, 3), (1, 6), (1, 11), (1, 4), (1, 10), (3, 6), (3, 11), (3, 4), (3, 10), (4, 6), (6, 11), (6, 10), (6, 9), (4, 11), (9, 11), (10, 11), (4, 8), (4, 10), (4, 9), (4, 5), (4, 7), (7, 10), (2, 9), (8, 9), (5, 8), (7, 8), (5, 7).
Question: What is the Jaccard coefficient between node 6 and node 7?
You need to format your answer as a float number. | 0.25 | graph_jaccard_coefficient |
The task is to determine the number of connected components in the graph.
A connected component is a subgraph where any two nodes are connected to each other by paths.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. The graph contains edges: (1, 11), (1, 6), (8, 11), (3, 6), (2, 8), (2, 7), (5, 7), (3, 4), (3, 5), (4, 9), (9, 10).
Question: How many connected components are there in the graph?
| 1 | graph_connected_component_number |
The task is to determine the Wiener index of a connected graph.
The Wiener index of a graph is the sum of the shortest-path distances between each pair of reachable nodes. For pairs of nodes in undirected graphs, only one orientation of the pair is counted.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, and 17. The graph contains edges: (1, 15), (12, 15), (15, 16), (2, 5), (2, 8), (5, 17), (5, 9), (4, 8), (7, 8), (3, 9), (9, 14), (4, 16), (4, 13), (10, 16), (13, 16), (16, 17), (11, 17), (6, 7), (10, 14), (11, 12).
Question: What is the Wiener index of the graph?
You need to format your answer as a float number. | 409.0 | graph_wiener_index |
The task is to determine the betweenness centrality of a node in the graph.
Betweenness centrality of a node u is the sum of the fraction of all-pairs shortest paths that pass through u.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, and 8. The graph contains edges: (1, 2), (1, 4), (1, 5), (1, 6), (1, 7), (2, 3), (2, 4), (2, 5), (4, 5), (4, 6), (4, 7), (5, 7), (6, 7), (6, 8).
Question: What is the betweenness centrality of node 4 in the graph?
You need to format your answer as a float number. | 0.1587301587301587 | graph_betweenness_centrality |
The task is to determine the Wiener index of a connected graph.
The Wiener index of a graph is the sum of the shortest-path distances between each pair of reachable nodes. For pairs of nodes in undirected graphs, only one orientation of the pair is counted.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, and 30. The graph contains edges: (1, 11), (1, 10), (1, 23), (1, 29), (1, 13), (1, 15), (1, 17), (1, 21), (1, 22), (1, 26), (1, 2), (1, 6), (1, 25), (1, 27), (6, 11), (8, 11), (9, 11), (10, 11), (11, 20), (11, 24), (11, 29), (11, 13), (11, 17), (11, 19), (11, 21), (11, 26), (11, 25), (11, 30), (2, 10), (6, 10), (10, 23), (10, 29), (10, 13), (10, 15), (10, 17), (10, 21), (10, 22), (10, 26), (10, 25), (10, 27), (2, 23), (3, 23), (13, 23), (15, 23), (18, 23), (21, 23), (22, 23), (23, 28), (23, 27), (6, 29), (8, 29), (9, 29), (13, 29), (17, 29), (19, 29), (20, 29), (21, 29), (24, 29), (25, 29), (26, 29), (29, 30), (2, 13), (6, 13), (13, 15), (13, 17), (13, 21), (13, 22), (13, 26), (13, 25), (13, 27), (2, 15), (3, 15), (15, 18), (15, 21), (15, 22), (15, 28), (15, 27), (6, 17), (8, 17), (9, 17), (17, 20), (17, 24), (17, 19), (17, 21), (17, 26), (17, 25), (17, 30), (2, 21), (6, 21), (21, 22), (21, 26), (21, 25), (21, 27), (2, 22), (3, 22), (18, 22), (22, 28), (22, 27), (6, 26), (8, 26), (9, 26), (19, 26), (20, 26), (24, 26), (25, 26), (26, 30), (2, 18), (2, 28), (2, 3), (2, 27), (6, 20), (6, 24), (6, 9), (6, 19), (6, 25), (6, 30), (6, 8), (8, 25), (9, 25), (19, 25), (20, 25), (24, 25), (25, 30), (3, 27), (18, 27), (27, 28), (3, 18), (18, 28), (3, 28), (4, 5), (4, 12), (4, 16), (4, 20), (4, 24), (4, 9), (4, 19), (4, 7), (4, 14), (4, 30), (4, 8), (5, 12), (5, 16), (5, 20), (5, 24), (5, 9), (5, 19), (5, 7), (5, 14), (5, 30), (5, 8), (7, 12), (8, 12), (9, 12), (12, 16), (12, 20), (12, 24), (12, 19), (12, 14), (12, 30), (7, 16), (8, 16), (9, 16), (14, 16), (16, 20), (16, 24), (16, 19), (16, 30), (7, 20), (8, 20), (9, 20), (14, 20), (19, 20), (20, 24), (20, 30), (7, 24), (8, 24), (9, 24), (14, 24), (19, 24), (24, 30), (7, 9), (8, 9), (9, 19), (9, 14), (9, 30), (7, 19), (8, 19), (14, 19), (19, 30), (7, 14), (7, 30), (7, 8), (8, 14), (14, 30), (8, 30).
Question: What is the Wiener index of the graph?
You need to format your answer as a float number. | 897.0 | graph_wiener_index |
The task is to determine the periphery of a graph.
The periphery of a graph is the set of nodes that are farthest from the center of the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24. The graph contains edges: (1, 2), (1, 6), (1, 8), (1, 7), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (6, 18), (6, 12), (6, 7), (3, 8), (8, 17), (8, 9), (3, 5), (3, 9), (4, 5), (4, 10), (4, 11), (4, 12), (5, 11), (5, 13), (5, 9), (9, 11), (9, 13), (9, 17), (10, 15), (10, 11), (10, 16), (11, 16), (11, 13), (11, 17), (13, 16), (13, 17), (13, 24), (13, 22), (13, 23), (18, 19), (16, 17), (22, 24), (23, 24), (21, 22), (22, 23), (21, 23), (14, 21), (19, 20).
Question: What is the periphery of the graph?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [14,20] | graph_periphery |
The task is to determine if the graph has a cycle.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16. The graph contains edges: (1, 6), (6, 10), (2, 3), (3, 14), (3, 16), (3, 4), (5, 15), (5, 13), (7, 15), (9, 10), (7, 9), (8, 11), (11, 12).
Question: Does the graph have a cycle?
Your answer should be Yes or No. | No | graph_has_cycle |
The task is to determine the degree centrality of a node in the graph.
The degree centrality values are normalized by dividing by the maximum possible degree in a graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, and 5. The graph contains edges: (1, 3), (1, 2), (2, 5), (4, 5).
Question: What is the degree centrality of node 5 in the graph?
You need to format your answer as a float number. | 0.5 | graph_degree_centrality |
The task is to determine the number of edges in the graph. For the undirected graph, you should count the edge between two nodes only once.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The graph contains edges: (1, 8), (1, 5), (2, 5), (2, 13), (10, 13), (3, 9), (3, 14), (9, 12), (4, 14), (10, 14), (11, 14), (14, 15), (4, 10), (4, 7), (6, 7), (11, 15).
Question: How many edges are there in the graph?
| 16 | graph_edge_number |
The task is to determine the global efficiency of a graph.
Global efficiency is the average of the inverse shortest path lengths between all pairs of nodes in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, and 31. The graph contains edges: (1, 2), (1, 3), (1, 5), (1, 7), (1, 8), (2, 3), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 9), (5, 9), (5, 10), (5, 11), (7, 8), (7, 14), (8, 15), (4, 10), (4, 11), (4, 12), (4, 6), (4, 13), (6, 13), (6, 29), (9, 10), (10, 11), (10, 12), (11, 13), (12, 28), (13, 29), (29, 30), (15, 26), (23, 28), (24, 28), (28, 30), (26, 27), (16, 17), (16, 18), (18, 25), (22, 25), (24, 25), (25, 27), (19, 20), (19, 21), (20, 22), (20, 23), (20, 21), (22, 23), (22, 24), (23, 24), (24, 30), (27, 31).
Question: What is the global efficiency of the graph?
You need to format your answer as a float number. | 0.3623169482846879 | graph_global_efficiency |
The task is to determine the density of the graph.
Density is defined as the ratio of the number of edges in the graph to the number of possible edges. For undirected graphs, the density is 2*m/(n*(n-1)), where m is the edge number and n is the node number. For directed graphs, the density is m/(n*(n-1)).
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24. The graph contains edges: (1, 18), (1, 19), (1, 20), (1, 21), (1, 9), (1, 22), (1, 11), (1, 7), (1, 8), (18, 19), (18, 20), (18, 21), (18, 22), (19, 20), (19, 21), (19, 22), (9, 20), (11, 20), (20, 21), (20, 22), (9, 21), (11, 21), (21, 22), (9, 11), (7, 22), (8, 22), (7, 11), (7, 8), (2, 17), (2, 3), (2, 4), (12, 17), (13, 17), (3, 5), (3, 6), (3, 4), (4, 5), (4, 6), (5, 6), (10, 12), (10, 13), (10, 14), (12, 13), (12, 14), (13, 14), (15, 16), (15, 23), (15, 24), (16, 23), (16, 24), (23, 24).
Question: What is the density of the graph?
You need to format your answer as a float number. | 0.18478260869565216 | graph_density |
The task is to determine the Wiener index of a connected graph.
The Wiener index of a graph is the sum of the shortest-path distances between each pair of reachable nodes. For pairs of nodes in undirected graphs, only one orientation of the pair is counted.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, and 20. The graph contains edges: (1, 4), (1, 7), (1, 9), (1, 15), (1, 19), (1, 20), (1, 2), (1, 3), (1, 5), (1, 8), (1, 12), (1, 10), (1, 16), (1, 17), (1, 18), (1, 6), (1, 11), (1, 13), (1, 14), (2, 4), (3, 4), (4, 7), (4, 9), (4, 15), (4, 19), (4, 20), (4, 5), (4, 8), (4, 12), (4, 10), (4, 16), (4, 17), (4, 18), (4, 6), (4, 11), (4, 13), (4, 14), (2, 7), (3, 7), (5, 7), (6, 7), (7, 9), (7, 15), (7, 19), (7, 20), (7, 8), (7, 12), (7, 10), (7, 16), (7, 17), (7, 18), (7, 11), (7, 13), (7, 14), (2, 9), (3, 9), (5, 9), (6, 9), (8, 9), (9, 15), (9, 19), (9, 20), (9, 12), (9, 10), (9, 16), (9, 17), (9, 18), (9, 11), (9, 13), (9, 14), (2, 15), (3, 15), (5, 15), (6, 15), (8, 15), (10, 15), (11, 15), (12, 15), (13, 15), (14, 15), (15, 19), (15, 20), (15, 16), (15, 17), (15, 18), (2, 19), (3, 19), (5, 19), (6, 19), (8, 19), (10, 19), (11, 19), (12, 19), (13, 19), (14, 19), (16, 19), (17, 19), (18, 19), (19, 20), (2, 20), (3, 20), (5, 20), (6, 20), (8, 20), (10, 20), (11, 20), (12, 20), (13, 20), (14, 20), (16, 20), (17, 20), (18, 20), (2, 3), (2, 5), (2, 8), (2, 12), (2, 10), (2, 16), (2, 17), (2, 18), (2, 6), (2, 11), (2, 13), (2, 14), (3, 5), (3, 8), (3, 12), (3, 10), (3, 16), (3, 17), (3, 18), (3, 6), (3, 11), (3, 13), (3, 14), (5, 8), (5, 12), (5, 10), (5, 16), (5, 17), (5, 18), (5, 6), (5, 11), (5, 13), (5, 14), (6, 8), (8, 12), (8, 10), (8, 16), (8, 17), (8, 18), (8, 11), (8, 13), (8, 14), (6, 12), (11, 12), (12, 16), (12, 17), (12, 18), (12, 13), (12, 14), (6, 10), (10, 16), (10, 17), (10, 18), (10, 11), (10, 13), (10, 14), (6, 16), (11, 16), (13, 16), (14, 16), (16, 17), (16, 18), (6, 17), (11, 17), (13, 17), (14, 17), (17, 18), (6, 18), (11, 18), (13, 18), (14, 18), (6, 11), (6, 13), (6, 14), (11, 13), (11, 14), (13, 14).
Question: What is the Wiener index of the graph?
You need to format your answer as a float number. | 191.0 | graph_wiener_index |
The task is to determine if the graph is Eulerian.
An Eulerian graph is a graph that contains an Eulerian circuit, which is a cycle that visits every edge exactly once.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. The graph contains edges: (1, 3), (1, 8), (2, 3), (6, 8), (7, 8), (2, 10), (10, 12), (4, 7), (5, 11), (9, 11).
Question: Is the graph Eulerian?
Your answer should be Yes or No. | No | graph_is_eularian |
The task is to determine the density of the graph.
Density is defined as the ratio of the number of edges in the graph to the number of possible edges. For undirected graphs, the density is 2*m/(n*(n-1)), where m is the edge number and n is the node number. For directed graphs, the density is m/(n*(n-1)).
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, and 6. The graph contains edges: (1, 6), (1, 2), (4, 6), (2, 3), (4, 5).
Question: What is the density of the graph?
You need to format your answer as a float number. | 0.3333333333333333 | graph_density |
The task is to determine the Adamic-Adar index of two nodes in a graph.
The Adamic-Adar index is the sum of the inverse logarithm of the degrees of the common neighbors of the two nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14. The graph contains edges: (1, 4), (1, 2), (1, 6), (4, 11), (4, 14), (2, 9), (6, 12), (9, 10), (3, 10), (3, 7), (7, 8), (11, 13), (5, 12).
Question: What is the Adamic-Adar index between node 2 and node 4?
You need to format your answer as a float number. | 0.9102392266268373 | graph_adamic_adar_index |
The task is to determine the barycenter of a graph.
The barycenter of a graph is also called the median. It includes the node that minimizes the sum of distances to all other nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13. The graph contains edges: (1, 8), (1, 6), (5, 8), (5, 6), (6, 12), (2, 5), (2, 9), (2, 13), (3, 5), (5, 12), (9, 12), (9, 11), (4, 13), (10, 13), (3, 4), (4, 11), (7, 10).
Question: What is the barycenter of the graph?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [2,5] | graph_barycenter |
The task is to determine the density of the graph.
Density is defined as the ratio of the number of edges in the graph to the number of possible edges. For undirected graphs, the density is 2*m/(n*(n-1)), where m is the edge number and n is the node number. For directed graphs, the density is m/(n*(n-1)).
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. The graph contains edges: (1, 11), (1, 9), (2, 10), (6, 10), (7, 10), (3, 8), (5, 8), (4, 12), (4, 7), (5, 12).
Question: What is the density of the graph?
You need to format your answer as a float number. | 0.15151515151515152 | graph_density |
The task is to determine the betweenness centrality of a node in the graph.
Betweenness centrality of a node u is the sum of the fraction of all-pairs shortest paths that pass through u.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, and 23. The graph contains edges: (1, 19), (1, 10), (18, 19), (9, 10), (2, 22), (2, 3), (4, 22), (4, 8), (6, 8), (5, 11), (11, 13), (7, 17), (12, 17), (17, 23), (9, 14), (14, 21), (13, 15), (12, 16), (15, 16), (18, 20).
Question: What is the betweenness centrality of node 9 in the graph?
You need to format your answer as a float number. | 0.04329004329004329 | graph_betweenness_centrality |
The task is to determine the Adamic-Adar index of two nodes in a graph.
The Adamic-Adar index is the sum of the inverse logarithm of the degrees of the common neighbors of the two nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, and 5. The graph contains edges: (1, 3), (1, 2), (2, 4), (4, 5).
Question: What is the Adamic-Adar index between node 2 and node 5?
You need to format your answer as a float number. | 1.4426950408889634 | graph_adamic_adar_index |
The task is to determine the degree of a node in the graph. For the undirected graph, you should count the edge between two nodes only once.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, and 32. The graph contains edges: (1, 29), (1, 5), (2, 29), (5, 29), (7, 29), (23, 29), (26, 29), (29, 30), (5, 30), (5, 24), (5, 6), (5, 11), (2, 7), (2, 20), (20, 25), (3, 24), (3, 11), (3, 4), (4, 24), (11, 24), (4, 11), (14, 30), (25, 30), (8, 16), (8, 17), (8, 9), (8, 10), (8, 12), (16, 26), (16, 17), (17, 26), (9, 10), (9, 12), (10, 12), (13, 15), (13, 19), (13, 27), (15, 31), (15, 22), (15, 25), (15, 32), (19, 27), (18, 31), (21, 31), (22, 31), (25, 31), (31, 32), (22, 25), (28, 32), (18, 28).
Question: What is the degree of node 17 in the graph?
| 3 | graph_degree |
The task is to determine the transitivity of a graph.
Transitivity is the ratio of the number of triangles in the graph to the number of connected triples of nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, and 5. The graph contains edges: (1, 3), (1, 4), (2, 4), (2, 5).
Question: What is the transitivity of the graph?
You need to format your answer as a float number. | 0 | graph_transitivity |
The task is to determine the Wiener index of a connected graph.
The Wiener index of a graph is the sum of the shortest-path distances between each pair of reachable nodes. For pairs of nodes in undirected graphs, only one orientation of the pair is counted.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and 18. The graph contains edges: (1, 13), (1, 4), (13, 15), (4, 16), (2, 10), (2, 9), (8, 10), (3, 18), (3, 11), (7, 18), (7, 11), (11, 12), (14, 16), (5, 6), (5, 17), (5, 12), (6, 14), (6, 17), (8, 12).
Question: What is the Wiener index of the graph?
You need to format your answer as a float number. | 725.0 | graph_wiener_index |
The task is to determine the topological sort of a directed acyclic graph (DAG).
Here is a directed graph. In the graph, (u, v) means that there is an edge from node u to node v. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, and 8. The graph contains edges: (1, 8), (1, 7), (8, 7), (7, 4), (2, 3), (3, 5), (5, 6), (6, 1).
Question: What is the topological sort of the directed acyclic graph (DAG)?
You need to format your answer as a list of nodes, e.g., [node-1, node-2, ..., node-n]. | [2,3,5,6,1,8,7,4] | graph_topological_sort |
The task is to determine the minimum spanning tree of a graph.
A minimum spanning tree is a subset of the edges that connects all vertices in the graph with the minimum possible total edge weight. If not specified, all edges have equal edge weights
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, and 17. The graph contains edges: (1, 2), (1, 3), (1, 5), (1, 6), (1, 7), (1, 10), (1, 9), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (3, 4), (3, 6), (3, 11), (3, 12), (5, 6), (5, 7), (5, 10), (4, 6), (6, 11), (6, 12), (10, 17), (8, 9), (9, 16), (9, 15), (4, 13), (4, 11), (4, 12), (11, 13), (11, 12), (12, 13), (8, 16), (8, 15), (14, 16), (15, 16), (14, 15).
Question: What is the minimum spanning tree of the graph?
You need to format your answer as a list of edges in ascending dictionary order, e.g., [(u1, v1), (u2, v2), ..., (un, vn)]. | [(1,2),(1,3),(1,5),(1,6),(1,7),(1,9),(1,10),(2,4),(3,11),(3,12),(4,13),(8,9),(8,15),(8,16),(10,17),(14,16)] | graph_minimum_spanning_tree |
The task is to determine the neighbors of a node in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, and 29. The graph contains edges: (1, 11), (1, 20), (11, 12), (11, 29), (2, 3), (2, 26), (3, 28), (7, 26), (4, 23), (4, 25), (18, 23), (5, 14), (5, 27), (12, 14), (16, 27), (6, 7), (7, 13), (8, 13), (9, 10), (10, 22), (17, 22), (15, 24), (21, 24), (17, 18), (19, 21).
Question: What are the neighbors of node 5 in the graph?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [14,27] | graph_neighbor |
The task is to determine the constraint of a node in a graph.
The constraint of a node is a measure of how much the node is constrained by its neighbors.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, and 9. The graph contains edges: (1, 2), (1, 7), (2, 5), (4, 7), (5, 7), (3, 5), (4, 5), (6, 8), (6, 9), (8, 9).
Question: What is the constraint of node 7 in the graph?
You need to format your answer as a float number. | 0.5347222222222222 | graph_constraint |
The task is to determine the number of strongly connected components in a directed graph.
A strongly connected component is a maximal subgraph where every node is reachable from every other node.
Here is a directed graph. In the graph, (u, v) means that there is an edge from node u to node v. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, and 25. The graph contains edges: (1, 4), (1, 3), (4, 1), (4, 14), (3, 1), (3, 24), (2, 12), (2, 21), (12, 15), (12, 19), (21, 2), (24, 3), (24, 6), (14, 4), (14, 25), (5, 19), (5, 23), (5, 13), (19, 12), (19, 5), (19, 17), (19, 21), (23, 5), (23, 17), (13, 5), (13, 17), (6, 24), (6, 10), (6, 11), (10, 20), (10, 6), (11, 20), (11, 6), (8, 16), (16, 8), (16, 18), (16, 22), (9, 18), (9, 22), (18, 16), (18, 9), (22, 16), (22, 9), (20, 10), (20, 11), (15, 12), (15, 7), (17, 19), (17, 23), (17, 13), (25, 14).
Question: How many strongly connected components are there in the graph?
| 4 | graph_strongly_connected_number |
The task is to determine the neighbors of a node in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, and 29. The graph contains edges: (1, 6), (1, 12), (1, 22), (1, 4), (1, 27), (1, 5), (1, 7), (4, 6), (6, 27), (6, 12), (6, 23), (2, 12), (3, 12), (4, 12), (5, 12), (9, 12), (11, 12), (12, 14), (12, 19), (12, 27), (12, 21), (12, 23), (12, 25), (12, 22), (12, 13), (12, 15), (4, 22), (5, 22), (8, 22), (9, 22), (16, 22), (21, 22), (22, 27), (22, 25), (22, 23), (4, 14), (4, 24), (4, 16), (4, 23), (4, 13), (4, 17), (4, 18), (3, 27), (13, 27), (14, 27), (16, 27), (17, 27), (18, 27), (21, 27), (23, 27), (3, 5), (5, 14), (5, 19), (5, 16), (5, 21), (5, 23), (5, 13), (5, 17), (5, 18), (5, 15), (5, 25), (7, 10), (3, 10), (3, 21), (3, 29), (3, 16), (3, 8), (14, 21), (15, 21), (16, 21), (17, 21), (18, 21), (20, 21), (21, 24), (8, 16), (9, 16), (14, 16), (15, 16), (16, 23), (16, 17), (8, 19), (14, 20), (14, 18), (24, 26), (24, 28), (13, 23), (17, 23), (18, 23), (17, 25), (17, 20), (9, 18), (18, 19), (15, 26), (15, 28), (9, 25), (9, 11).
Question: What are the neighbors of node 29 in the graph?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [3] | graph_neighbor |
The task is to determine the degree centrality of a node in the graph.
The degree centrality values are normalized by dividing by the maximum possible degree in a graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, and 33. The graph contains edges: (1, 2), (2, 24), (12, 24), (3, 32), (3, 22), (4, 32), (9, 22), (10, 22), (11, 22), (21, 22), (22, 23), (22, 26), (4, 10), (4, 11), (10, 18), (9, 11), (11, 15), (5, 14), (5, 18), (5, 9), (9, 14), (9, 26), (9, 21), (6, 20), (6, 21), (12, 20), (16, 20), (21, 33), (21, 26), (7, 25), (7, 28), (8, 25), (8, 28), (12, 28), (17, 28), (23, 28), (28, 29), (8, 31), (30, 31), (16, 26), (19, 26), (12, 15), (15, 19), (15, 30), (12, 19), (19, 29), (13, 23), (13, 27), (17, 33).
Question: What is the degree centrality of node 18 in the graph?
You need to format your answer as a float number. | 0.0625 | graph_degree_centrality |
Given nodes u and v in a graph, the task is to determine if there is an edge connecting the two nodes.
For undirected graphs, it means whether there are edges between u and v. For directed graphs, it means whether there are edges from u to v.Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. The graph contains edges: (1, 11), (1, 12), (7, 12), (8, 12), (2, 9), (2, 5), (2, 6), (2, 3), (2, 4), (3, 9), (4, 9), (5, 9), (6, 9), (9, 10), (3, 5), (4, 5), (5, 10), (5, 6), (3, 6), (4, 6), (6, 10), (3, 4), (7, 8).
Question: Is there an edge between node 6 and node 12?
Your answer should be Yes or No. | No | graph_edge_existence |
The task is to determine the closeness centrality of a node in the graph.
For a node u, closeness centrality is the reciprocal of the average shortest path distance to u over all n-1 reachable nodes. For directed graphs, it computes the incoming distance to u.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, and 29. The graph contains edges: (1, 22), (1, 15), (1, 12), (1, 13), (12, 22), (13, 22), (12, 15), (12, 13), (13, 16), (13, 17), (2, 11), (2, 3), (2, 23), (2, 4), (2, 10), (2, 26), (3, 11), (4, 11), (10, 11), (11, 23), (3, 23), (3, 4), (3, 10), (3, 26), (4, 23), (10, 23), (23, 26), (4, 10), (4, 26), (10, 26), (5, 6), (5, 28), (6, 19), (6, 28), (19, 20), (7, 16), (7, 17), (16, 17), (8, 9), (8, 29), (9, 27), (9, 24), (9, 29), (24, 29), (24, 27), (14, 21), (14, 20), (14, 18), (14, 25), (18, 21), (20, 21), (18, 20), (20, 25), (18, 25).
Question: What is the closeness centrality of node 10 in the graph?
You need to format your answer as a float number. | 0.21428571428571427 | graph_closeness_centrality |
The task is to determine the resource allocation index of two nodes in a graph.
The resource allocation index of two nodes is the sum of the inverse of the degrees of the common neighbors of the two nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, and 23. The graph contains edges: (1, 4), (1, 14), (1, 16), (4, 12), (9, 14), (2, 8), (2, 9), (2, 22), (8, 10), (17, 22), (19, 22), (3, 17), (3, 7), (17, 19), (7, 15), (12, 13), (5, 6), (6, 11), (6, 21), (6, 18), (6, 20), (10, 15), (15, 19), (19, 23).
Question: What is the resource allocation index between node 5 and node 20?
You need to format your answer as a float number. | 0.2 | graph_resource_allocation_index |
The task is to determine the harmonic centrality of a node in the graph.
Harmonic centrality of a node u is the sum of the reciprocal of the shortest path distances from all other nodes to u (direction is not considered for undirected graphs).
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and 18. The graph contains edges: (1, 15), (1, 3), (1, 18), (1, 10), (1, 11), (2, 15), (3, 15), (6, 15), (7, 15), (9, 15), (10, 15), (15, 18), (15, 16), (3, 9), (3, 13), (3, 18), (3, 11), (7, 18), (11, 18), (16, 18), (9, 10), (2, 16), (4, 13), (4, 17), (4, 6), (6, 17), (5, 6), (6, 14), (6, 8), (5, 12).
Question: What is the harmonic centrality of node 11 in the graph?
You need to format your answer as a float number. | 8.2 | graph_harmonic_centrality |
The task is to determine if there is a triangle in the graph that includes a specific node.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, and 8. The graph contains edges: (1, 8), (1, 5), (5, 6), (2, 3), (3, 4), (4, 7), (6, 7).
Question: Is there a triangle in the graph that includes node 6?
Your answer should be Yes or No. | No | graph_triangle_detection |
The task is to determine the number of connected components in the graph.
A connected component is a subgraph where any two nodes are connected to each other by paths.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, and 19. The graph contains edges: (1, 3), (1, 19), (1, 5), (1, 6), (3, 19), (3, 5), (3, 6), (6, 19), (5, 15), (5, 6), (6, 14), (6, 15), (2, 12), (2, 13), (7, 12), (10, 12), (11, 12), (12, 16), (12, 18), (12, 13), (7, 13), (10, 13), (11, 13), (13, 16), (13, 18), (4, 9), (14, 15), (15, 17), (14, 17), (7, 10), (7, 16), (7, 18), (8, 10), (10, 11), (8, 11).
Question: How many connected components are there in the graph?
| 3 | graph_connected_component_number |
The task is to determine if the graph is Eulerian.
An Eulerian graph is a graph that contains an Eulerian circuit, which is a cycle that visits every edge exactly once.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, and 5. The graph contains edges: (1, 2), (1, 5), (1, 3), (1, 4), (2, 5), (2, 3), (2, 4), (3, 5), (4, 5), (3, 4).
Question: Is the graph Eulerian?
Your answer should be Yes or No. | Yes | graph_is_eularian |
The task is to determine the diameter of a graph.
The diameter of a graph is the longest shortest path between any two nodes in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and 18. The graph contains edges: (1, 5), (1, 2), (1, 9), (2, 5), (4, 5), (5, 16), (5, 10), (5, 9), (5, 8), (2, 16), (2, 10), (2, 9), (2, 8), (2, 17), (2, 14), (3, 16), (6, 16), (11, 16), (13, 16), (16, 18), (7, 10), (10, 15), (10, 17), (10, 14), (10, 12).
Question: What is the diameter of the graph?
| 4 | graph_diameter |
The task is to determine the neighbors of a node in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. The graph contains edges: (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 8), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (3, 5), (3, 6), (3, 8), (3, 9), (4, 5), (4, 6), (4, 7), (5, 6), (5, 7), (5, 10), (5, 11), (5, 12), (6, 7), (6, 10), (6, 11), (8, 9), (7, 11), (10, 11), (10, 12), (11, 12).
Question: What are the neighbors of node 6 in the graph?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [1,2,3,4,5,7,10,11] | graph_neighbor |
The task is to determine the Wiener index of a connected graph.
The Wiener index of a graph is the sum of the shortest-path distances between each pair of reachable nodes. For pairs of nodes in undirected graphs, only one orientation of the pair is counted.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, and 9. The graph contains edges: (1, 9), (1, 6), (4, 9), (2, 6), (3, 8), (3, 4), (4, 8), (5, 8), (5, 7).
Question: What is the Wiener index of the graph?
You need to format your answer as a float number. | 105.0 | graph_wiener_index |
The task is to determine the betweenness centrality of a node in the graph.
Betweenness centrality of a node u is the sum of the fraction of all-pairs shortest paths that pass through u.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, and 31. The graph contains edges: (1, 3), (1, 5), (3, 6), (5, 9), (5, 6), (2, 30), (4, 30), (27, 30), (29, 30), (6, 12), (4, 20), (4, 31), (20, 23), (23, 31), (29, 31), (8, 9), (9, 12), (7, 10), (7, 26), (10, 11), (10, 25), (25, 26), (11, 24), (21, 25), (24, 25), (22, 24), (13, 14), (15, 17), (15, 16), (17, 18), (16, 18), (18, 19), (21, 22), (27, 28), (28, 29).
Question: What is the betweenness centrality of node 7 in the graph?
You need to format your answer as a float number. | 0.0019157088122605363 | graph_betweenness_centrality |
The task is to determine the radius of a graph.
The radius of a graph is the minimum eccentricity of any node in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, and 28. The graph contains edges: (1, 25), (1, 28), (3, 25), (25, 28), (3, 28), (27, 28), (2, 7), (2, 12), (2, 23), (7, 12), (7, 16), (7, 26), (12, 15), (12, 16), (4, 23), (14, 23), (3, 21), (3, 27), (13, 21), (19, 21), (10, 27), (17, 27), (18, 27), (4, 14), (4, 10), (9, 10), (5, 15), (15, 16), (6, 11), (6, 8), (11, 13), (8, 17), (8, 18), (9, 26), (24, 26), (9, 17), (17, 18), (9, 18), (18, 24), (13, 20), (19, 20), (19, 22).
Question: What is the radius of the graph?
| 6 | graph_radius |
The task is to determine common neighbors between two nodes in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. The graph contains edges: (1, 8), (1, 9), (1, 2), (1, 10), (6, 8), (7, 8), (8, 9), (8, 10), (2, 9), (6, 9), (9, 10), (2, 5), (2, 10), (7, 10), (4, 5), (3, 4), (6, 7).
Question: What are the common neighbors between node 9 and node 6?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [8] | graph_common_neighbor |
The task is to determine the number of nodes in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, and 9. The graph contains edges: (1, 5), (1, 9), (1, 3), (3, 5), (4, 5), (5, 6), (5, 9), (3, 9), (2, 3), (2, 8), (7, 8), (4, 6).
Question: How many nodes are there in the graph?
| 9 | graph_node_number |
The task is to determine the maximal independent set guaranteed to contain a given node in the graph.
An independent set is a set of nodes such that the subgraph induced by these nodes contains no edges. A maximal independent set is an independent set such that it is not possible to add a new node and still get an independent set.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, and 8. The graph contains edges: (1, 2), (1, 4), (2, 7), (4, 8), (5, 7), (3, 8), (5, 6).
Question: What is the maximal independent set that includes node 4 of the graph?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [2,3,4,6] | graph_maximal_independent_set |
The task is to determine the resource allocation index of two nodes in a graph.
The resource allocation index of two nodes is the sum of the inverse of the degrees of the common neighbors of the two nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, and 5. The graph contains edges: (1, 2), (2, 3), (3, 5), (4, 5).
Question: What is the resource allocation index between node 2 and node 5?
You need to format your answer as a float number. | 0.5 | graph_resource_allocation_index |
The task is to determine the global efficiency of a graph.
Global efficiency is the average of the inverse shortest path lengths between all pairs of nodes in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. The graph contains edges: (1, 5), (1, 10), (2, 5), (8, 10), (2, 9), (3, 9), (3, 11), (4, 11), (6, 8), (7, 12).
Question: What is the global efficiency of the graph?
You need to format your answer as a float number. | 0.30741943241943237 | graph_global_efficiency |
The task is to find all bridges of a graph.
A bridge is an edge in a graph whose removal increases the number of connected components.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, and 25. The graph contains edges: (1, 9), (1, 10), (1, 8), (8, 9), (9, 11), (8, 10), (10, 20), (10, 24), (8, 23), (8, 11), (8, 22), (2, 3), (2, 6), (2, 11), (3, 25), (3, 4), (3, 5), (3, 6), (3, 7), (4, 6), (6, 25), (11, 23), (11, 22), (4, 25), (5, 25), (4, 5), (4, 7), (5, 7), (7, 15), (7, 16), (15, 16), (22, 23), (12, 20), (18, 20), (19, 20), (20, 24), (12, 24), (21, 24), (12, 19), (18, 19), (13, 14), (13, 17), (14, 17), (17, 21).
Question: What are the bridges of the graph?
You need to format your answer as a list of edges in ascending dictionary order, e.g., [(u1, v1), (u2, v2), ..., (un, vn)]. | [(2,11),(17,21),(21,24)] | graph_bridges |
The task is to determine the barycenter of a graph.
The barycenter of a graph is also called the median. It includes the node that minimizes the sum of distances to all other nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, and 28. The graph contains edges: (1, 2), (1, 4), (1, 5), (1, 7), (1, 8), (1, 9), (2, 3), (2, 4), (2, 5), (2, 6), (3, 4), (4, 5), (4, 7), (4, 9), (4, 11), (3, 5), (5, 7), (5, 6), (7, 8), (7, 9), (7, 26), (8, 9), (8, 24), (8, 25), (8, 26), (9, 24), (9, 25), (9, 26), (3, 10), (3, 6), (3, 11), (3, 12), (3, 13), (6, 10), (6, 12), (6, 13), (10, 12), (10, 13), (11, 28), (11, 20), (11, 21), (11, 12), (12, 20), (12, 13), (23, 26), (24, 26), (25, 26), (23, 24), (24, 25), (23, 25), (20, 28), (21, 28), (15, 20), (15, 21), (21, 27), (14, 15), (14, 16), (14, 18), (14, 19), (14, 17), (15, 17), (16, 18), (16, 19), (18, 22), (18, 19), (19, 22).
Question: What is the barycenter of the graph?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [11] | graph_barycenter |
The task is to determine the periphery of a graph.
The periphery of a graph is the set of nodes that are farthest from the center of the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, and 22. The graph contains edges: (1, 2), (1, 5), (1, 9), (1, 10), (1, 11), (1, 12), (1, 14), (1, 15), (1, 16), (2, 5), (2, 6), (2, 7), (2, 9), (2, 10), (2, 11), (2, 12), (2, 14), (2, 15), (2, 16), (2, 19), (2, 22), (2, 4), (2, 20), (2, 3), (2, 21), (2, 13), (3, 5), (4, 5), (5, 6), (5, 7), (5, 9), (5, 10), (5, 11), (5, 12), (5, 14), (5, 15), (5, 16), (5, 19), (5, 22), (5, 17), (5, 20), (5, 8), (5, 21), (5, 13), (5, 18), (3, 9), (4, 9), (6, 9), (7, 9), (9, 10), (9, 11), (9, 12), (9, 14), (9, 15), (9, 16), (9, 19), (9, 22), (9, 17), (9, 20), (9, 21), (9, 18), (3, 10), (6, 10), (7, 10), (8, 10), (10, 11), (10, 12), (10, 14), (10, 15), (10, 16), (10, 19), (10, 22), (10, 17), (10, 21), (10, 13), (10, 18), (6, 11), (7, 11), (8, 11), (11, 12), (11, 14), (11, 15), (11, 16), (11, 19), (11, 22), (11, 17), (11, 20), (11, 21), (11, 13), (11, 18), (4, 12), (6, 12), (7, 12), (8, 12), (12, 14), (12, 15), (12, 16), (12, 19), (12, 22), (12, 17), (12, 20), (12, 13), (12, 18), (3, 14), (6, 14), (7, 14), (8, 14), (13, 14), (14, 15), (14, 19), (14, 17), (14, 20), (14, 21), (14, 18), (4, 15), (6, 15), (7, 15), (13, 15), (15, 16), (15, 22), (15, 17), (15, 21), (15, 18), (3, 16), (6, 16), (7, 16), (16, 17), (16, 20), (16, 18), (3, 6), (4, 6), (6, 7), (6, 19), (6, 22), (6, 8), (6, 13), (6, 18), (3, 7), (4, 7), (7, 19), (7, 22), (7, 17), (7, 20), (7, 8), (4, 19), (19, 20), (19, 21).
Question: What is the periphery of the graph?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [1,2,3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22] | graph_periphery |
The task is to determine the number of strongly connected components in a directed graph.
A strongly connected component is a maximal subgraph where every node is reachable from every other node.
Here is a directed graph. In the graph, (u, v) means that there is an edge from node u to node v. The graph contains nodes: 1, 2, 3, 4, 5, and 6. The graph contains edges: (1, 2), (1, 4), (2, 1), (2, 3), (4, 1), (4, 3), (3, 6), (3, 5), (6, 3), (5, 3).
Question: How many strongly connected components are there in the graph?
| 2 | graph_strongly_connected_number |
The task is to determine the number of nodes in the graph.
Here is a directed graph. In the graph, (u, v) means that there is an edge from node u to node v. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24. The graph contains edges: (1, 2), (1, 8), (1, 5), (2, 1), (2, 6), (2, 7), (2, 9), (2, 10), (2, 12), (2, 16), (2, 3), (2, 21), (2, 23), (8, 15), (8, 19), (5, 1), (5, 7), (5, 9), (5, 11), (5, 14), (5, 16), (5, 17), (5, 3), (5, 18), (5, 20), (5, 22), (7, 24), (3, 2), (3, 5), (3, 13), (4, 2), (4, 5).
Question: How many nodes are there in the graph?
| 24 | graph_node_number |
The task is to determine the minimum vertex cover of a graph.
A vertex cover is a set of nodes such that every edge in the graph is incident to at least one node in the set.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. The graph contains edges: (1, 10), (1, 5), (3, 10), (10, 11), (10, 12), (2, 3), (2, 4), (2, 6), (2, 7), (3, 4), (3, 7), (3, 8), (3, 9), (4, 7), (4, 8), (4, 9), (6, 7), (8, 9), (11, 12).
Question: What is the minimum vertex cover of the graph?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [1,2,3,4,7,8,10,11] | graph_min_vertex_cover |
The task is to determine the average degree of the neighbors of a node in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, and 22. The graph contains edges: (1, 18), (2, 18), (3, 18), (5, 18), (6, 18), (4, 7), (4, 11), (4, 9), (9, 22), (8, 17), (8, 16), (8, 13), (12, 17), (14, 17), (15, 16), (13, 21), (13, 19), (12, 22), (10, 21), (10, 20).
Question: What is the average neighbor degree of node 20 in the graph?
You need to format your answer as a float number. | 2.0 | graph_avg_neighbor_degree |
The task is to determine if the graph is Eulerian.
An Eulerian graph is a graph that contains an Eulerian circuit, which is a cycle that visits every edge exactly once.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, and 7. The graph contains edges: (1, 5), (1, 7), (1, 2), (1, 3), (5, 6), (4, 7), (6, 7), (2, 4).
Question: Is the graph Eulerian?
Your answer should be Yes or No. | No | graph_is_eularian |
The task is to determine if there is a triangle in the graph that includes a specific node.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, and 9. The graph contains edges: (1, 5), (1, 9), (1, 3), (3, 5), (4, 5), (5, 6), (5, 9), (3, 9), (2, 3), (2, 8), (7, 8), (4, 6).
Question: Is there a triangle in the graph that includes node 5?
Your answer should be Yes or No. | Yes | graph_triangle_detection |
The task is to determine the minimum edge covering of a graph.
An edge cover is a set of edges such that every vertex in the graph is incident to at least one edge in the set. The minimum edge cover is the edge cover with the smallest number of edges.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, and 23. The graph contains edges: (1, 11), (5, 11), (8, 11), (10, 11), (11, 12), (11, 15), (2, 20), (2, 9), (3, 20), (16, 20), (5, 9), (9, 21), (3, 16), (6, 16), (16, 22), (4, 6), (4, 22), (6, 22), (13, 22), (21, 22), (5, 10), (5, 15), (10, 17), (8, 15), (7, 19), (18, 19), (19, 23), (8, 12), (8, 14), (14, 17), (13, 21), (18, 23).
Question: What is the minimum edge covering of the graph?
You need to format your answer as a list of edges in ascending dictionary order, e.g., [(u1, v1), (u2, v2), ..., (un, vn)]. | [(3,20),(6,16),(9,2),(11,1),(12,11),(13,21),(14,8),(15,5),(17,10),(19,7),(22,4),(23,18)] | graph_min_edge_covering |
The task is to determine the local connectivity of two nodes in the graph.
Local connectivity is whether there exists at least one path between the two nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and 18. The graph contains edges: (1, 18), (1, 2), (1, 12), (1, 11), (2, 18), (3, 18), (4, 18), (7, 18), (11, 18), (12, 18), (13, 18), (2, 12), (2, 11), (3, 12), (4, 12), (7, 12), (11, 12), (12, 13), (3, 14), (3, 13), (3, 10), (3, 4), (3, 8), (3, 7), (4, 14), (5, 14), (6, 14), (7, 14), (8, 14), (10, 14), (13, 14), (14, 15), (4, 13), (7, 13), (8, 13), (10, 13), (4, 10), (5, 10), (6, 10), (7, 10), (8, 10), (10, 15), (4, 8), (4, 7), (5, 8), (6, 8), (7, 8), (8, 15), (5, 6), (5, 16), (5, 15), (6, 16), (6, 15), (15, 16), (16, 17), (9, 17).
Question: What is the local connectivity between node 12 and node 11 in the graph?
| Yes | graph_local_connectivity |
The task is to determine if the directed graph is strongly connected.
A directed graph is strongly connected if there is a directed path between every pair of vertices.
Here is a directed graph. In the graph, (u, v) means that there is an edge from node u to node v. The graph contains nodes: 1, 2, 3, 4, and 5. The graph contains edges: (2, 1), (3, 5), (5, 4), (4, 2).
Question: Is the directed graph strongly connected?
Your answer should be Yes or No. | No | graph_is_strongly_connected |
The task is to determine the neighbors of a node in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, and 9. The graph contains edges: (1, 3), (1, 5), (1, 4), (1, 6), (4, 5), (5, 9), (5, 6), (4, 9), (2, 7), (2, 8), (7, 9).
Question: What are the neighbors of node 7 in the graph?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [2,9] | graph_neighbor |
The task is to determine the periphery of a graph.
The periphery of a graph is the set of nodes that are farthest from the center of the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, and 6. The graph contains edges: (1, 6), (1, 3), (1, 5), (3, 4), (3, 5), (2, 4).
Question: What is the periphery of the graph?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [2,6] | graph_periphery |
The task is to determine the Wiener index of a connected graph.
The Wiener index of a graph is the sum of the shortest-path distances between each pair of reachable nodes. For pairs of nodes in undirected graphs, only one orientation of the pair is counted.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, and 9. The graph contains edges: (1, 4), (1, 3), (1, 2), (4, 9), (4, 8), (3, 8), (3, 6), (2, 7), (5, 7), (7, 8), (5, 8), (5, 6), (5, 9).
Question: What is the Wiener index of the graph?
You need to format your answer as a float number. | 64.0 | graph_wiener_index |
The task is to determine the number of nodes in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, and 22. The graph contains edges: (1, 5), (1, 11), (1, 19), (1, 2), (2, 5), (3, 5), (5, 21), (5, 12), (5, 14), (5, 11), (5, 20), (5, 9), (5, 13), (10, 11), (2, 19), (3, 19), (10, 19), (12, 19), (13, 19), (14, 19), (16, 19), (17, 19), (18, 19), (19, 21), (19, 20), (2, 12), (2, 18), (2, 3), (2, 10), (2, 9), (2, 13), (12, 21), (12, 16), (12, 15), (3, 18), (17, 18), (18, 21), (3, 21), (3, 16), (3, 14), (4, 10), (7, 10), (10, 21), (10, 17), (9, 21), (9, 20), (6, 13), (13, 21), (14, 21), (20, 21), (4, 14), (8, 14), (17, 20), (20, 22).
Question: How many nodes are there in the graph?
| 22 | graph_node_number |
The task is to determine if the directed graph is strongly connected.
A directed graph is strongly connected if there is a directed path between every pair of vertices.
Here is a directed graph. In the graph, (u, v) means that there is an edge from node u to node v. The graph contains nodes: 1, 2, 3, 4, and 5. The graph contains edges: (1, 2), (1, 4), (2, 1), (4, 1), (4, 5), (3, 5), (5, 3), (5, 4).
Question: Is the directed graph strongly connected?
Your answer should be Yes or No. | Yes | graph_is_strongly_connected |
The task is to determine the number of nodes in the graph.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. The graph contains edges: (1, 2), (1, 11), (2, 12), (4, 12), (6, 12), (3, 9), (3, 5), (9, 10), (7, 8), (8, 10).
Question: How many nodes are there in the graph?
| 12 | graph_node_number |
The task is to determine the number of isolated nodes in the graph.
An isolated node is a node that has no edges connecting it to any other nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, and 31. The graph contains edges: (1, 14), (1, 23), (1, 12), (1, 15), (1, 25), (1, 30), (2, 14), (3, 14), (4, 14), (5, 14), (7, 14), (10, 14), (12, 14), (14, 18), (14, 17), (14, 23), (14, 25), (14, 21), (14, 31), (14, 24), (14, 26), (2, 23), (3, 23), (4, 23), (5, 23), (6, 23), (8, 23), (9, 23), (10, 23), (11, 23), (12, 23), (13, 23), (15, 23), (16, 23), (17, 23), (18, 23), (21, 23), (23, 28), (23, 31), (23, 25), (23, 26), (23, 27), (23, 29), (23, 30), (3, 12), (12, 28), (12, 21), (12, 31), (12, 25), (12, 29), (12, 26), (12, 15), (2, 15), (3, 15), (4, 15), (5, 15), (7, 15), (10, 15), (11, 15), (15, 17), (15, 18), (15, 21), (15, 19), (15, 31), (15, 25), (15, 26), (2, 25), (3, 25), (4, 25), (5, 25), (7, 25), (8, 25), (9, 25), (10, 25), (11, 25), (13, 25), (16, 25), (17, 25), (18, 25), (22, 25), (24, 25), (25, 31), (25, 26), (25, 27), (25, 29), (25, 30), (28, 30), (2, 31), (2, 28), (2, 11), (2, 21), (4, 31), (6, 31), (9, 31), (16, 31), (17, 31), (18, 31), (22, 31), (26, 31), (27, 31), (29, 31), (4, 28), (7, 28), (9, 28), (10, 28), (13, 28), (16, 28), (18, 28), (26, 28), (3, 11), (9, 11), (10, 11), (11, 17), (11, 18), (11, 26), (11, 27), (11, 16), (11, 21), (6, 21), (7, 21), (8, 21), (9, 21), (10, 21), (13, 21), (16, 21), (17, 21), (18, 21), (21, 26), (21, 27), (21, 29), (3, 17), (3, 7), (3, 9), (3, 18), (3, 26), (3, 27), (3, 16), (3, 29), (3, 6), (3, 8), (5, 17), (10, 17), (16, 17), (17, 22), (17, 26), (17, 27), (17, 29), (7, 29), (7, 26), (7, 8), (5, 9), (4, 18), (6, 18), (18, 22), (4, 26), (5, 26), (20, 26), (22, 26), (5, 27), (20, 27), (5, 16), (16, 22), (6, 29), (4, 6), (4, 8), (8, 24), (5, 22).
Question: How many isolated nodes are there in the graph?
| 0 | graph_isolate_number |
The task is to determine the number of strongly connected components in a directed graph.
A strongly connected component is a maximal subgraph where every node is reachable from every other node.
Here is a directed graph. In the graph, (u, v) means that there is an edge from node u to node v. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24. The graph contains edges: (1, 13), (13, 18), (13, 1), (13, 6), (2, 19), (2, 21), (19, 7), (19, 2), (21, 2), (21, 6), (3, 11), (11, 3), (11, 23), (11, 6), (4, 14), (4, 22), (14, 4), (22, 3), (22, 4), (5, 20), (5, 9), (20, 24), (20, 5), (9, 24), (9, 5), (6, 13), (6, 16), (6, 21), (6, 11), (16, 18), (16, 6), (7, 19), (7, 15), (15, 18), (15, 7), (8, 10), (10, 8), (10, 12), (24, 20), (24, 9), (12, 10), (12, 17), (23, 17), (23, 11), (17, 23), (17, 12), (18, 13), (18, 15), (18, 16).
Question: How many strongly connected components are there in the graph?
| 3 | graph_strongly_connected_number |
The task is to determine the minimum spanning tree of a graph.
A minimum spanning tree is a subset of the edges that connects all vertices in the graph with the minimum possible total edge weight. If not specified, all edges have equal edge weights
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, and 19. The graph contains edges: (1, 10), (1, 11), (3, 10), (4, 10), (5, 10), (10, 11), (10, 13), (10, 16), (11, 13), (11, 16), (2, 15), (2, 12), (2, 8), (8, 15), (12, 15), (14, 15), (8, 12), (12, 19), (12, 14), (8, 14), (3, 16), (3, 4), (3, 5), (3, 6), (13, 16), (4, 5), (5, 6), (7, 13), (14, 19), (9, 17), (9, 19), (9, 18).
Question: What is the minimum spanning tree of the graph?
You need to format your answer as a list of edges in ascending dictionary order, e.g., [(u1, v1), (u2, v2), ..., (un, vn)]. | [(1,10),(1,11),(2,8),(2,12),(2,15),(3,4),(3,5),(3,6),(3,10),(3,16),(7,13),(8,14),(9,17),(9,18),(9,19),(10,13),(12,19)] | graph_minimum_spanning_tree |
The task is to determine the barycenter of a graph.
The barycenter of a graph is also called the median. It includes the node that minimizes the sum of distances to all other nodes.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, and 5. The graph contains edges: (1, 3), (1, 5), (2, 3), (4, 5).
Question: What is the barycenter of the graph?
You need to format your answer as a list of nodes in ascending order, e.g., [node-1, node-2, ..., node-n]. | [1] | graph_barycenter |
The task is to determine the minimum spanning tree of a graph.
A minimum spanning tree is a subset of the edges that connects all vertices in the graph with the minimum possible total edge weight. If not specified, all edges have equal edge weights
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14. The graph contains edges: (1, 2), (1, 7), (2, 11), (2, 14), (7, 8), (5, 11), (11, 13), (13, 14), (3, 5), (4, 6), (6, 12), (10, 12), (8, 9).
Question: What is the minimum spanning tree of the graph?
You need to format your answer as a list of edges in ascending dictionary order, e.g., [(u1, v1), (u2, v2), ..., (un, vn)]. | [(1,2),(1,7),(2,11),(2,14),(3,5),(4,6),(5,11),(6,12),(7,8),(8,9),(10,12),(11,13)] | graph_minimum_spanning_tree |
The task is to determine the closeness centrality of a node in the graph.
For a node u, closeness centrality is the reciprocal of the average shortest path distance to u over all n-1 reachable nodes. For directed graphs, it computes the incoming distance to u.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, and 26. The graph contains edges: (1, 8), (1, 17), (3, 8), (8, 9), (8, 13), (17, 23), (2, 15), (11, 15), (15, 21), (4, 19), (19, 25), (5, 18), (5, 11), (5, 7), (6, 18), (10, 18), (10, 11), (6, 7), (7, 26), (7, 10), (12, 26), (16, 26), (10, 12), (14, 23), (21, 22), (16, 24), (22, 25), (20, 22).
Question: What is the closeness centrality of node 3 in the graph?
You need to format your answer as a float number. | 0.10315789473684212 | graph_closeness_centrality |
The task is to determine the minimum edge covering of a graph.
An edge cover is a set of edges such that every vertex in the graph is incident to at least one edge in the set. The minimum edge cover is the edge cover with the smallest number of edges.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, and 32. The graph contains edges: (1, 13), (1, 31), (13, 30), (13, 21), (2, 24), (2, 3), (17, 24), (21, 24), (3, 21), (21, 22), (4, 14), (14, 19), (14, 23), (5, 17), (6, 7), (6, 8), (7, 20), (8, 26), (15, 20), (26, 30), (9, 10), (9, 16), (9, 25), (10, 28), (11, 16), (16, 18), (16, 23), (12, 25), (28, 29), (11, 12), (19, 27), (22, 32), (29, 32).
Question: What is the minimum edge covering of the graph?
You need to format your answer as a list of edges in ascending dictionary order, e.g., [(u1, v1), (u2, v2), ..., (un, vn)]. | [(2,24),(3,2),(7,6),(12,11),(14,4),(15,20),(17,5),(18,16),(22,21),(23,14),(25,9),(26,8),(27,19),(28,10),(29,32),(30,13),(31,1)] | graph_min_edge_covering |
The task is to determine the minimum edge covering of a graph.
An edge cover is a set of edges such that every vertex in the graph is incident to at least one edge in the set. The minimum edge cover is the edge cover with the smallest number of edges.
Here is an undirected graph. In the graph, (u, v) means that node u and node v are connected. The graph contains nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, and 17. The graph contains edges: (1, 17), (1, 12), (12, 16), (2, 16), (2, 6), (2, 4), (4, 6), (3, 9), (3, 8), (3, 13), (5, 9), (7, 9), (8, 9), (5, 11), (5, 7), (7, 11), (10, 14), (10, 15), (14, 15).
Question: What is the minimum edge covering of the graph?
You need to format your answer as a list of edges in ascending dictionary order, e.g., [(u1, v1), (u2, v2), ..., (un, vn)]. | [(1,17),(2,16),(3,8),(4,6),(5,9),(7,11),(10,14),(12,1),(13,3),(15,10)] | graph_min_edge_covering |
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