# Iterated line graphs with only negative
eigenvalues $-2$, their complements and energy

Harishchandra S. Ramanea,, B. Parvathalub,
Daneshwari Patila, K. Ashokaa Corresponding author

###### Abstract

The graphs with all equal negative or positive eigenvalues are special kind in
the spectral graph theory. In this article, several iterated line graphs
$\mathcal{L}^{k}(G)$ with all equal negative eigenvalues $-2$ are
characterized for $k\geq 1$ and their energy consequences are presented. Also,
the spectra and the energy of complement of these graphs are obtained,
interestingly they have exactly two positive eigenvalues with different
multiplicities. Moreover, we characterize a large class of equienergetic
graphs which generalize some of the existing results. There are two different
quotient matrices defined for an equitable partition of $H$-join (generalized
composition) of regular graphs to find the spectrum (partial) of adjacency
matrix, Laplacian matrix and signless Laplacian matrix, it has been proved
that these two quotient matrices give the same respective spectrum of graphs.

Keywords: Eigenvalues of graphs, Energy of a graph, Equitable partition,
Iterated line graphs, Complement of iterated line graphs.

Mathematics Subject Classification: 05C50, 05C76.

aDepartment of Mathematics, Karnatak University, Dharwad - 580003,
Karnataka, India

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
bDepartment of Mathematics, Karnatak University’s Karnatak Arts College,
Dharwad - 580001, Karnataka, India

<EMAIL_ADDRESS>
##  1 Introduction

Spectral graph theory is aimed at answering the questions related to the graph
theory in terms of eigenvalues of matrices which are naturally associated with
graphs. Graphs with all equal positive or negative eigenvalues are very
special kind in the study of eigenvalues of graphs and its easy to find energy
on these class of graphs. Also, the graphs with an exact number of positive or
negative eigenvalues has applications in finding equienergetic graphs. The
line graph of a graph has a special property that its least eigenvalue is not
smaller than $-2$. In the last two decades graphs with least eigenvalues $-2$
have been well studied [9, 10, 39]. The problem of particular interest in this
class of graphs is the graphs with all negative eigenvalues equal to $-2$.
This problem can be restated in terms of the eigenvalues of signless Laplacian
matrix Q for line graphs as the graphs with all its signless Laplacian
eigenvalues belongs to the set $[2,\infty)\cup\\{0\\}$ with the help of the
relation (1). In [33] Ramane et al. obtained the spectra and the energy of
iterated regular line graphs $\mathcal{L}^{k}(G)$ for $k\geq 2$ and presented
infinitely many pairs of non-trivial equienergetic graphs which belong to the
above class of graphs. Let $\varrho$ be the property that a graph $G$ has all
its negative eigenvalues equal to $-2$. In this paper, we are motivated to
find several classes of iterated line graphs $\mathcal{L}^{k}(G)$ with the
property $\varrho$ for $k\geq 1$, spectra of their complements and energy
consequences. As a consequence of energy of these graphs we present a large
class of equienergetic graphs which generalize some of the existing results.
The energy relation between a regular graph $G$ and its complement were
studied in [27, 28, 29, 30, 32], we extend the results pertaining
$\mathcal{E}(\overline{G})=\mathcal{E}(G)$ to non-regular iterated line graphs
with the property $\rho$. The energy of line graphs is well studied in [11,
18, 24, 33, 34]. In this paper, we also present hyperenergetic iterated line
graphs and their complements. The software sage math [37] is used to verify
some of the results.
This paper is organized as follows. In section 2, basic definitions, equitable
partition, known results on spectra and energy of graphs are presented. In
section 3, it is proved that the two different quotient matrices defined for
an equitable partition of $H$-join of regular graphs give the same
spectrum(partial) for adjacency matrix, Laplacian matrix and signless
Laplacian matrix. Section 4 provides the results on spectra and energy of
iterated line graphs satisfying the property $\rho$. Section 5 deals with the
spectra and the energy of complements of graphs which are presented in section
4. In section 6, upper bound is given for independence number of iterated line
graphs and their complements. Also, a result on minimum order of the connected
graph when its complement is connected and line graph satisfying the property
$\rho$.

##  2 Preliminaries

In this paper, simple and undirected graphs $G$ are considered with vertex set
$V(G)=\\{v_{1},v_{2},\ldots,v_{n}\\}$ and edge set
$E(G)=\\{e_{1},e_{2},\ldots,e_{m}\\}$. Let $e=v_{i}v_{j}$ be an edge of $G$
with its end vertices $v_{i}$ and $v_{j}$. The order and size of a graph $G$
is the number of vertices and number of edges respectively in it. The
complement $\overline{G}$ of a graph $G$ has same vertices as in $G$ but two
vertices are adjacent in $\overline{G}$ if and only if they are not adjacent
in $G$. Let the subgraph of a graph $G$ obtained by deleting vertices
$v_{1},v_{2},\ldots,v_{k}$, $k<n$ and the edges incident to them in $G$ be
$G-\\{v_{1},v_{2},\ldots,v_{k}\\}$ and simply $G-v$ if one vertex $v$ and
edges incident to it are deleted. The adjacency matrix $A(G)$ or simply $A$ of
a graph $G$ of order $n$ is the $n\times n$ matrix indexed by $V(G)$ whose
$(i,j)$-th entry is defined as $a_{ij}=1$ if $v_{i}v_{j}\in E(G)$ and $0$
otherwise. The Laplacian matrix $L$ and signless Laplacian matrix $Q$ of a
graph is defined as $L=D-A$ and $Q=D+A$ respectively, where $D=[d_{i}]$ is the
diagonal degree matrix of suitable order whose $i$-th diagonal entry $d_{i}$
is the degree of the vertex $v_{i}$. The matrix $Q$ is positive semi-definite
and has real eigenvalues. Given any square matrix $M$, the multiset of its
eigenvalues is called the spectrum of $M$. Denote the characteristic
polynomial and spectrum of the matrix $M$, respectively by $\varphi(M;x)$ and
$Sp(M)$. Let $aSp(M)\pm b=\\{a\lambda\pm b:\lambda\in Sp(M)\\}$ for any two
real numbers $a$ and $b$. The spectrum of a graph is the spectrum of its
adjacency matrix $A$. The positive (negative) inertia of a graph $G$ is the
number of positive (negative) eigenvalues of $G$ denoted by $n^{+}(n^{-})$.
Given a graph $G$ we denote the spectrum of adjacency matrix and signless
Laplacian matrix respectively by
$Sp_{A}(G)=\\{\lambda_{1}^{m_{1}},\lambda_{2}^{m_{2}},\ldots,\lambda_{k}^{m_{k}}\\}$
and $Sp_{Q}(G)=\\{q_{1}^{m_{1}},q_{2}^{m_{2}},\ldots,q_{k}^{m_{k}}\\}$, where
$\lambda_{i}$’s and $q_{i}$’s are indexed in descending order, and $m_{i}$ is
the multiplicity of the respective eigenvalue for $1\leq i\leq k$. Denote the
least eigenvalue of signless Laplacian by $q_{\min}$. The energy [16] of a
graph $G$ is defined as

$\mathcal{E}(G)=\sum\limits_{i=1}^{n}\lvert\lambda_{i}\rvert=2\sum\limits_{i=1}^{n^{+}}\lambda_{i}=-2\sum\limits_{i=1}^{n^{-}}\lambda_{n-i+1}.$

Two graphs $G_{1}$ and $G_{2}$ of same order are said to be equienergetic if
their energies are equal. A set of vertices in a graph $G$ is independent if
no two vertices in the set are adjacent. The independence number $\alpha{(G)}$
of a graph $G$ is the maximum cardinality of an independent set of vertices in
$G$. As usual the graphs $C_{n},K_{n}$ and $P_{n}$ denote the cycle, complete
graph and path on $n$ vertices respectively.

######  Definition 2.1.

[26] The Turán graph $T_{r}(n)$, $r\geq 1$, is the complete $r$-partite graph
of order $n$ with all parts of size either $\lfloor n/r\rfloor$ or $\lceil
n/r\rceil$.

######  Definition 2.2.

[20] The line graph $\mathcal{L}(G)$ of a graph $G$ is the graph with vertex
set same as the edge set of $G$, two vertices in $\mathcal{L}(G)$ are adjacent
if the corresponding edges in $G$ have a vertex in common. The $k$-th iterated
line graph of $G$ for $k\in\\{0,1,2,\ldots\\}$ is defined as
$\mathcal{L}^{k}(G)=\mathcal{L}(\mathcal{L}^{k-1}(G))$, where
$\mathcal{L}^{0}(G)=G$ and $\mathcal{L}^{1}(G)=\mathcal{L}(G)$.

Let $n_{k}$ and $m_{k}$ denote the order and the size of $\mathcal{L}^{k}(G)$
for $k\in\\{0,1,2,\ldots\\}$. It is noted that $n_{k}=m_{k-1}$ for
$k\in\\{1,2,\ldots\\}$. Let us denote the eigenvalues of $\mathcal{L}^{k}(G)$
and its complement $\overline{\mathcal{L}^{k}(G)}$ respectively by
$\lambda_{k(j)},\overline{\lambda}_{k(j)}$ for $k\in\\{1,2,\ldots\\}$ and
$1\leq j\leq n_{k}$. The complement of a line graph is also called jump graph
[6]. The important spectral property of line graph is that its least
eigenvalue is not less than $-2$ [22].

######  Theorem 2.3.

[8] If $G$ is a $r$-regular graph of order $n$ with spectrum
$Sp_{A}(G)=\\{r,\lambda_{2},\ldots,\lambda_{n}\\}$, then the complement graph
of $G$ is a $(n-r-1)$-regular graph with spectrum
$Sp_{A}(\overline{G})=\\{n-r-1,-1-\lambda_{n},\ldots,-1-\lambda_{2}\\}$.

######  Theorem 2.4.

[35] If $G$ is a $r$-regular graph of order $n$ and size $m$ with spectrum
$Sp_{A}(G)=\\{r,\lambda_{2},\ldots,\lambda_{n}\\}$, then line graph of $G$ is
a $(2r-2)$-regular graph with spectrum
$Sp_{A}(\mathcal{L}(G))=\\{2r-2,\lambda_{2}+r-2,\ldots,\lambda_{n}+r-2,-2^{m-n}\\}$.

Let $n$ and $m$ be the order and the size of a graph $G$. The following is the
relation between the eigenvalues of line graph and signless Laplacian of a
graph $G$ [7]

$Sp_{A}\big{(}\mathcal{L}(G)\big{)}=\\{-2^{m-n}\\}\cup\big{(}Sp_{Q}(G)-2\big{)}.$
(1)

the multiplicity of the eigenvalue $-2$ in $\mathcal{L}(G)$ is equal to
$m-n+1$ if $G$ is bipartite and equal to $m-n$ if $G$ is not bipartite [9].

######  Theorem 2.5.

[11] Let $G$ be a graph of order $n>2$ with $m$ edges and minimum degree
$\delta$. If $\delta\geq\frac{n}{2}+1$. Then the line graph $\mathcal{L}(G)$
satisfies the property $\rho$ with energy $4(m-n)$. Thus, the line graphs of
all such graphs of order $n$ and size $m$ with property
$\delta\geq\frac{n}{2}+1$ are mutually equienergetic.

######  Theorem 2.6.

[24] Let $G$ be a graph of order $n\geq 5$ and size $m$. If $m\geq 2n$, the
line graph $\mathcal{L}(G)$ is hyperenergetic.

The following is an elegant relation even though not sharp between the
smallest eigenvalue $\lambda_{min}$ of $A$ and the smallest eigenvalue
$q_{min}$ of $Q$.

######  Proposition 2.7.

[14] If $G$ is a graph with minimum degree $\delta$ and maximum degree
$\Delta$, then

$q_{min}-\Delta\leq\lambda_{min}\leq q_{min}-\delta.$

######  Proposition 2.8.

[13] If $G$ is a spanning subgraph of a graph $G^{\prime}$, then
$q_{min}(G)\leq q_{min}(G^{\prime})$.

######  Theorem 2.9.

[21] If $G$ is a graph with vertex $v$, then $q_{min}(G)-1\leq q_{min}(G-v)$.

Let ${\bf j}$ denotes all one’s column vector.

######  Theorem 2.10.

[19] If $\lambda\in Sp_{A}(G)$, then $-1-\lambda\in Sp_{A}(\overline{G})$ if
and only if ${\bf j}^{T}Y=0$ for some eigenvector $Y$ of $G$ corresponding to
the eigenvalue $\lambda$.

######  Proposition 2.11.

[8] In the line graph $\mathcal{L}(G)$ of a graph $G$ the eigenspace of the
eigenvalue $-2$ is orthogonal to the vector ${\bf j}$.

The Weyl’s eigenvalue inequality [23]
$\lambda_{j}(M_{1})+\lambda_{k}(M_{2})\leq\lambda_{i}(M_{1}+M_{2})$,
$j+k-n\geq i$ for sum of two Hermitian matrices $M_{1}$ and $M_{2}$ of order
$n$ gives the following useful eigenvalues inequality on a graph $G$ of order
$n$ and its complement $\overline{G}$ [27].

$\lambda_{j}+\overline{\lambda}_{n-j+2}\leq-1\text{ for
}j\in\\{2,3,\ldots,n\\}.$ (2)

######  Proposition 2.12.

[15] Let $M$ be any square matrix of order $n$ with the characteristic
polynomial $\varphi(M;x)=\sum\limits_{r=0}^{n}(-1)^{n}m_{r}x^{n-r}$. Then
$m_{r}$ is equal to the sum of the principal minors of $M$ of order $r$.

######  Definition 2.13.

[2] Let $G$ be a graph with vertex set $V(G)=\\{v_{1},v_{2}$, $\ldots$,
$v_{n}\\}$. The extended bipartite double $Ebd(G)$ of a graph $G$ is the
bipartite graph with its partite sets $X=\\{x_{1},x_{2},\ldots,x_{n}\\}$ and
$Y=\\{y_{1},y_{2},\ldots,y_{n}\\}$ in which two vertices $x_{i}$ and $y_{j}$
are adjacent iff $i=j$ or $v_{i}$ and $v_{j}$ are adjacent in $G$. The
$Ebd(G)$ is also called the extended double cover [1].

######  Theorem 2.14.

[2] Let $Ebd(G)$ is the extended bipartite double of a graph $G$.

  1. 1.

If $G$ is connected, then $Ebd(G)$ is connected.

  2. 2.

The spectrum of $Ebd(G)$ is $Sp_{A}(Ebd(G))=(-Sp_{A}(G)-1)\cup(Sp_{A}(G)+1)$.

######  Definition 2.15.

[36] If $G$ is a graph of order $n$ with labeled vertices, then the graph
$G[H_{1},H_{2}$,…, $H_{n}]$ called generalized composition or $H$-join [4]
which is obtained from the graphs $H_{1},H_{2},$ …, $H_{n}$ by joining every
vertex of $H_{i}$ to every vertex of $H_{j}$ if $v_{i}$ is adjacent to $v_{j}$
in $G$.

Let $\pi=(\pi_{1},\pi_{2},\ldots,\pi_{p})$ be a partition of a vertex set of a
graph $G$. The partition $\pi$ is called equitable partition [5, 36] if for
each $i,j=1,2,\ldots,p$ there exists a number $c_{ij}$ such that for every
vertex $v\in\pi_{i}$ there are exactly $c_{ij}$ edges between $v$ and the
vertices in $\pi_{j}$. If $\pi$ is an equitable partition, then the associated
$p\times p$ matrix with rows and columns corresponding to the partite sets
$\pi_{1},\pi_{2},\ldots,\pi_{p}$ is called quotient matrix. Let $A_{\pi}$ is
the $p\times p$ matrix with $(ij)$-th element $a^{\pi}_{ij}$ equal to
$c_{ij}$, and the $p\times p$ diagonal matrix $D_{\pi}$ whose $i$-th diagonal
element equal to $\sum\limits_{k=1}^{p}c_{ik}$. If $\pi$ is an equitable
partition, we denote the quotient matrix for adjacency, Laplacian and signless
Laplacian matrices of a graph, respectively by $A_{\pi},L_{\pi}$ and
$Q_{\pi}$. The matrices $A_{\pi},L_{\pi}$ and $Q_{\pi}$ are given by
$A_{\pi}=[a^{\pi}_{ij}]$ [36], $L_{\pi}=[l^{\pi}_{ij}]=D_{\pi}-A_{\pi}$ [3]
and $Q_{\pi}=[q^{\pi}_{ij}]=D_{\pi}+A_{\pi}$ [5, 40]. It is noted that these
are non-symmetric matrices.

Let $H_{i}$ be $r_{i}$-regular graphs of order $n_{i}$ for $i=1,2,\ldots,p$.
In case of $H$-join $G[H_{1},H_{2}$,…, $H_{p}]$, where $G$ is a graph of order
$p$, let us denote the quotient matrix for adjacency, Laplacian and signless
Laplacian of a graph, respectively by
$A^{H}_{\pi}=[a^{H}_{ij}],L^{H}_{\pi}=[l^{H}_{ij}]$ and
$Q^{H}_{\pi}=[q^{H}_{ij}]$. These matrices are defined as [4, 41]

$a^{H}_{ij}=\begin{dcases}c_{ij}&\text{ if }{i=j}\\\ \sqrt{n_{i}n_{j}}&\text{
if }{v_{i}v_{j}}\in E(G)\\\ 0&\text{ otherwise}\end{dcases},$
$l^{H}_{ij}=\begin{dcases}l^{\pi}_{ij}&\text{ if }{i=j}\\\
-\sqrt{n_{i}n_{j}}&\text{ if }{v_{i}v_{j}}\in E(G)\\\ 0&\text{
otherwise}\end{dcases}\ \text{ and }\
q^{H}_{ij}=\begin{dcases}q^{\pi}_{ij}&\text{ if }{i=j}\\\
\sqrt{n_{i}n_{j}}&\text{ if }{v_{i}v_{j}}\in E(G)\\\ 0&\text{
otherwise}\end{dcases}.$

It is noted that these are symmetric matrices.

If $\pi=(\pi_{1},\pi_{2},\ldots,\pi_{p})$ is an equitable partition of a graph
$G$ with cardinality of $\pi_{i}$, $\lvert\pi_{i}\rvert=m_{i}$ for
$i=1,\ldots,p$, then $\pi$ also equitable partition to its complement
$\overline{G}$. The quotient matrix $\overline{A}_{\pi}$ of $\overline{G}$ is
given by $\overline{A}_{\pi}={\bf J}_{\pi}-I-A_{\pi}$ [38], where ${\bf
J}_{\pi}$ is the matrix of order $p$ whose $(i,j)$-th element equal to $m_{j}$
and $I$ is the identity matrix of order $p$.

######  Proposition 2.16.

[38] Let the graphs $G_{1}$ and $G_{2}$ are co-spectral with equitable
partitions ${}^{1}\pi$ and ${}^{2}\pi$ respectively. If
$A_{{}^{1}\pi}=A_{{}^{2}\pi}$, then the graphs $\overline{G_{1}}$ and
$\overline{G_{2}}$ are co-spectral.

######  Theorem 2.17.

[41] Let $G$ be a labeled graph of order $n$ and $H_{i}$ be $r_{i}$-regular
graph of order $n_{i}$ for $i=1,2,\ldots,n$. If $\mathsf{G}=G[H_{1},H_{2}$,…,
$H_{n}]$, then

$Sp_{Q}(\mathsf{G})=\left(\cup_{i=1}^{n}\Big{(}(q^{\pi}_{ii}-2r_{i})+(Sp_{Q}(H_{i})\setminus\\{2r_{i}\\})\Big{)}\right)\cup\Big{(}Sp(Q^{H}_{\pi})\Big{)}.$

Suppose that $n\geq 2$ and $M=[m_{ij}]\in\mathbb{C}^{n\times n}$. The
Geršgorin discs $D_{i},i=1,2,\ldots,n$ of the matrix $M$ are defined as the
closed circular regions $D_{i}=\\{z\in\mathbb{C}:\lvert z-m_{ii}\rvert\leq
R_{i}\\}$ in the complex plane, where

$R_{i}=\sideset{}{}{\sum}_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\lvert m_{ij}\rvert\ \text{ is the radius of $D_{i}$. }$

######  Theorem 2.18 (Geršgorin).

[23] Let $n\geq 2$ and $M\in\mathbb{C}^{n\times n}$. All eigenvalues of the
matrix $M$ lie in the region $D=\cup_{i=1}^{n}D_{i}$, where
$D_{i},i=1,2,\ldots,n$ are the Geršgorin discs of $M$.

##  3 Spectra of Quotient Matrices

######  Theorem 3.1.

Let ${\bf G}=G[H_{1},H_{2}$,…, $H_{n}]$, where $H_{i}$ is $r_{i}$-regular
graph of order $n_{i}$ for $i=1,2,\ldots,n$. Then the spectrum of the quotient
matrices $A_{\pi},L_{\pi}$ and $Q_{\pi}$ equal to the spectrum of the quotient
matrices $A^{H}_{\pi},L^{H}_{\pi}$ and $Q^{H}_{\pi}$ respectively.

###### Proof.

It can be easily seen that $\pi=(V(H_{1}),V(H_{2}),\ldots,V(H_{n}))$ is an
equitable partition of ${\bf G}$. Let us first prove the result for the
matrices $Q_{\pi}$ and $Q^{H}_{\pi}$ with the help of Proposition 2.12 and
similar can be applied for the remaining matrices. The entries of the matrices
$Q_{\pi}$ and $Q^{H}_{\pi}$ can be written as $q^{\pi}_{ij}=a_{ij}n_{j}$,
$q^{H}_{ij}=a_{ij}\sqrt{n_{i}n_{j}}$ for $i\neq j$ and
$q^{\pi}_{ii}=q^{H}_{ii}$, where $a_{ij}$ is the entry of the adjacency matrix
of $G$. Let $S_{n}$ be the set of all permutations over the set
$\\{1,2,\ldots,n\\}$ and if $\sigma\in S_{n}$ denote its sign by
$sgn(\sigma)$. Let $M_{r}^{{}^{\prime}}$ be the principal minor of order $r$
which is obtained by deleting any $n-r$ rows and corresponding columns of the
matrix $Q_{\pi}$ and the respective principal minor of the matrix
$Q^{H}_{\pi}$ be $M_{r}^{{}^{\prime\prime}}$ for $1\leq r\leq n$.

$\displaystyle M_{r}^{{}^{\prime}}$ $\displaystyle=\sum\limits_{\sigma\in
S_{r}}sgn(\sigma)\prod\limits_{k=1}^{r}q^{\pi}_{k\sigma(k)}=\sum\limits_{\sigma\in
S_{r}}sgn(\sigma)\prod\limits_{k=1}^{r}a_{k\sigma(k)}n_{\sigma(k)}$
$\displaystyle=\sum\limits_{\sigma\in
S_{r}}sgn(\sigma)\prod\limits_{k=1}^{r}a_{k\sigma(k)}n_{k}\text{\qquad and }$
$\displaystyle M_{r}^{{}^{\prime\prime}}$
$\displaystyle=\sum\limits_{\sigma\in
S_{r}}sgn(\sigma)\prod\limits_{k=1}^{r}q^{H}_{k\sigma(k)}=\sum\limits_{\sigma\in
S_{r}}sgn(\sigma)\prod\limits_{k=1}^{r}a_{k\sigma(k)}\sqrt{n_{k}n_{\sigma(k)}}$
$\displaystyle=\sum\limits_{\sigma\in
S_{r}}sgn(\sigma)\prod\limits_{k=1}^{r}a_{k\sigma(k)}n_{k}\text{\quad if
}k\neq\sigma(k)\text{ for any }k=1,2,\ldots,r.$

If $k=\sigma(k)$ for $k=1,2,\ldots,p$ and $1\leq p\leq r$, then we have

$M_{r}^{{}^{\prime}}=M_{r}^{{}^{\prime\prime}}=\sum\limits_{\sigma\in
S_{r}}sgn(\sigma)\prod\limits_{k=1}^{r-p}a_{k\sigma(k)}n_{k}(q^{\pi}_{kk})^{p}.$

Which shows that $M_{r}^{{}^{\prime}}=M_{r}^{{}^{\prime\prime}}$ to each $r$
for $1\leq r\leq n$. Hence, the sum of principal minors of order $r$ of the
matrices $Q_{\pi}$ and $Q^{H}_{\pi}$ are equal. It shows that the matrices
$Q_{\pi}$ and $Q^{H}_{\pi}$ have the same characteristic polynomials by
Proposition 2.12. Thus, the matrices $Q_{\pi}$ and $Q^{H}_{\pi}$ have the same
spectrum. ∎

Now Theorem 2.17, with the help of above Theorem can be stated in the
following way.

######  Proposition 3.2.

Let $G$ be a labeled graph of order $n$ and $H_{i}$ be $r_{i}$-regular graph
of order $n_{i}$ for $i=1,2,\ldots,n$. If $\mathsf{G}=G[H_{1},H_{2}$,…,
$H_{n}]$, then

$Sp_{Q}(\mathsf{G})=\left(\cup_{i=1}^{n}\Big{(}R_{i}+(Sp_{Q}(H_{i})\setminus\\{2r_{i}\\})\Big{)}\right)\cup\Big{(}Sp(Q_{\pi})\Big{)}$

where $R_{i}$ is the $i$-th row sum excluding the diagonal entry of $Q_{\pi}$
for $i=1,2,\ldots,n$.

###### Proof.

Proof follows by Theorem 3.1 and an observation that
$q^{\pi}_{ii}=2r_{i}+R_{i}$ for $i=1,2,\ldots,n$. ∎

##  4 Spectra and Energy of iterated line graphs with the property $\rho$

######  Theorem 4.1.

Let $G$ be a graph of order $n_{0}$ and size $m_{0}$ with $d_{u}+d_{v}\geq 6$
to each edge $e=uv$ in $G$. Then the graphs $\mathcal{L}^{k}(G)$ satisfy the
property $\varrho$ for $k\geq 2$. And all the iterated line graphs
$\mathcal{L}^{k}(G)$ of such graphs $G$ are mutually equienergetic with energy
$4(n_{k}-n_{k-1})$ for $k\geq 2$.

###### Proof.

If $G$ is a graph of order $n_{0}$ and size $m_{0}$ with an edge $e=uv$, then
the degree of a vertex corresponding to an edge $e$ in $\mathcal{L}(G)$ is
$d_{u}+d_{v}-2$. But the condition $d_{u}+d_{v}\geq 6$ to each edge $e=uv$ in
$G$ implies $d_{u}+d_{v}-2\geq 4$ which shows that the minimum degree of each
vertex in $\mathcal{L}(G)$ is at least four. It is well known that the least
eigenvalue of the line graph $\mathcal{L}(G)$ is not smaller than $-2$. Hence,
the least eigenvalues $\lambda_{min},q_{min}$ and the minimum degree $\delta$
of $\mathcal{L}(G)$ by Proposition 2.7 satisfy
$q_{min}\geq\lambda_{min}+\delta\geq-2+4=2$. Now, by relation (1)
$\mathcal{L}(\mathcal{L}(G))=\mathcal{L}^{2}(G)$ satisfies the property $\rho$
with the multiplicity of $-2$ equal to $m_{1}-n_{1}$. One can find easily that
the minimum degree $\delta$ in the line graphs $\mathcal{L}^{k}(G)$ increases
as $k$ increases, which shows that $q_{min}$ of $\mathcal{L}^{k}(G)$ increases
as $k$ increases and which is at least $2$. Hence, by relation (1) the
iterated line graphs $\mathcal{L}^{k}(G)$ satisfy the property $\rho$ to each
$k\geq 2$ with their energy equal to $4(m_{k-1}-n_{k-1})=4(n_{k}-n_{k-1})$. ∎

A tree graph is called caterpillar if the removal of all pendant vertices in
it makes a path graph. The spectra and energy of line graph of caterpillars is
studied in [34]

######  Example 4.2.

The graph in Figure 1 is the caterpillar $C(4,3,4)$. This graph satisfies the
conditions in Theorem 4.1, therefore all the graphs
$\mathcal{L}^{k}\big{(}C(4,3,4)\big{)}$ satisfy the property $\rho$ for $k\geq
2$.

######  Corollary 4.3.

Let $G$ be a graph of order $n_{0}$ and size $m_{0}$ with minimum degree
$\delta\geq 3$. Then the graphs $\mathcal{L}^{k}(G)$ satisfy the property
$\varrho$ for $k\geq 2$. And all the iterated line graphs $\mathcal{L}^{k}(G)$
of such graphs $G$ are mutually equienergetic with energy $4(n_{k}-n_{k-1})$
for $k\geq 2$.

######  Remark 4.4.

In [33] Ramane et al. obtained the spectra and energy of iterated line graphs
of regular graphs of degree $r\geq 3$ and thereby characterized large class of
pairs of non-trivial equienergetic regular graphs. It is noted that the
results of [33] become particular case of Corollary 4.3.

Now, Theorem 4.1 naturally motivates to discuss when $\mathcal{L}(G)$
satisfies the property $\rho$ as $\mathcal{L}^{k}(G)$ satisfy the property
$\rho$ for $k\geq 2$ in a graph $G$ with $d_{u}+d_{v}\geq 6$ to each edge
$e=uv$. In this direction we have the following results.

######  Lemma 4.5.

Let $G$ be a labeled connected graph of order $n\geq 2$ and $H_{i}$ be
$r_{i}$-regular graph of order $n_{i}$, where $r_{i}\geq 1$, for
$i=1,2,\ldots,n$. Then the least eigenvalue of the quotient matrix $Q_{\pi}$
of the graph $G[H_{1},H_{2}$,…, $H_{n}]$ is at least 2.

###### Proof.

The signless Laplacian $Q$ of a graph $G$ is positive semi-definite and has
real eigenvalues which implies that the eigenvalues of quotient matrix
$Q_{\pi}$ are real by Proposition 3.2. Observe that in $i$-th row of $Q_{\pi}$
the diagonal entry $q^{\pi}_{ii}$ satisfies the equality
$q^{\pi}_{ii}=2r_{i}+R_{i}$, where $R_{i}$ is the $i$-th row sum excluding
$q^{\pi}_{ii}$ of $Q_{\pi}$ for $i=1,2,\ldots,n$. By using Geršgorin Theorem,
all the eigenvalues of $Q_{\pi}$ belong to the union of the closed intervals
$[2r_{i},2(r_{i}+R_{i})]$ for $i=1,2,\ldots,n$. Now, the result follows from
the condition that $r_{i}\geq 1$. ∎

######  Theorem 4.6.

Let $G$ be a labeled connected graph of order $n\geq 2$ and $H_{i}$ be
$r_{i}$-regular graph of order $p_{i}$, where $r_{i}\geq 1$, for
$i=1,2,\ldots,n$. If $\mathsf{G}=G[H_{1},H_{2}$,…, $H_{n}]$ is a graph of
order $n_{0}$ and size $m_{0}$, then the graphs $\mathcal{L}^{k}(\mathsf{G})$
satisfy the property $\rho$ for $k\geq 1$. And the line graphs of graphs
$G^{\prime}$ of order $n_{k}$ and size $m^{\prime}_{k}$ for which
$\mathcal{L}^{k}(\mathsf{G})$ is a spanning subgraph are mutually
equienergetic with energy $4(m^{\prime}_{k}-n_{k})$ for $k\geq 0$.

###### Proof.

As $G$ is connected graph of order $n\geq 2$, for every vertex $v_{i}$ in $G$
there exists at least one adjacent vertex $v_{j}$ which shows that each vertex
of the graph $H_{i}$ is adjacent to every vertex of $H_{j}$ for at least one
$j$ in $\mathsf{G}=G[H_{1},H_{2}$,…, $H_{n}]$. But $H_{i}$ are $r_{i}$-regular
graph of order $p_{i}$ with $r_{i}\geq 1$ implies $p_{i}\geq 2$. Thus, the
quotient matrix $Q_{\pi}$ corresponding to an equitable partition
$\pi=(V(H_{1}),V(H_{2}),\ldots,V(H_{n}))$ has non-diagonal entry
$q^{\pi}_{ij}\geq 2$ in its $i$-th row for at least one $j$. Since, the
signless Laplacian matrix $Q$ of any graph is positive semi-definite, it has
all non-negative eigenvalues. By using Proposition 3.2 and Lemma 4.5, we get
that every signless Laplacian eigenvalue of $\mathsf{G}$ is at least $2$. Now
by relation (1), the line graph $\mathcal{L}(\mathsf{G})$ satisfies the
property $\rho$. It can be observed that the minimum order graph $\mathsf{G}$
with possible least degrees is $\mathsf{G}=K_{2}[K_{2},K_{2}]$, which shows
that minimum degree $\delta$ of $\mathsf{G}$ is at least $3$. Thus, by
Corollary 4.3 all the iterated line graphs $\mathcal{L}^{k}(\mathsf{G})$ for
$k\geq 1$ satisfy the property $\rho$. Since, $\mathsf{G}$ is a spanning
subgraph of $G^{\prime}$, from Proposition 2.8 $G^{\prime}$ also has $q_{min}$
at least $2$ which implies that the graph $\mathcal{L}(G^{\prime})$ also
satisfies the property $\rho$ with its energy equal to
$4(m^{\prime}_{k}-n_{k})$ for $k\geq 0$ by relation (1). This completes the
proof. ∎

######  Remark 4.7.

Let $G$ be a labeled connected graph of order $n\geq 2$ and $H_{i}$ be
$r_{i}$-regular graph of order $p_{i}$ with $r_{i}\geq 1$, for
$i=1,2,\ldots,n$. Let $\mathsf{G}=G[H_{1}$, …, $H_{j},\ldots,H_{n}]$ for
$1\leq j\leq i$ and $H_{s_{1}}$, $H_{s_{2}}$ be two $r$-regular graphs of same
order with $r\geq 1$. If $\mathsf{G_{1}}=G[H_{1}$, …,
$H_{s_{1}},\ldots,H_{n}]$ and $\mathsf{G_{2}}=G[H_{1}$, …,
$H_{s_{2}},\ldots,H_{n}]$, then the graphs $\mathcal{L}^{k}(\mathsf{G_{1}})$
and $\mathcal{L}^{k}(\mathsf{G_{2}})$ are equienergetic for $k\geq 1$. And
also if $H_{s_{1}}$ and $H_{s_{2}}$ are non co-spectral (co-spectral) graphs,
then we get non co-spectral (co-spectral) graphs
$\mathcal{L}^{k}(\mathsf{G_{1}})$ and $\mathcal{L}^{k}(\mathsf{G_{2}})$
respectively for $k\geq 0$ as they have same quotient matrices.

######  Remark 4.8.

In Theorem 2.5, K. C. Das et al. characterized large class of equienergetic
line graphs of graphs of order $n$ with the condition on minimum degree
$\delta\geq\frac{n}{2}+1$. But one can construct equienergetic line graphs of
graphs of order $n$ with the condition $\delta\leq\frac{n}{2}+1$ by using
Theorem 4.6. For example if $\mathsf{G}=K_{2}[K_{2},C_{n}]$, $n\geq 6$ we can
always construct non-isomorphic equienergetic line graphs of graph
$\mathsf{G}$ with energy equal to $4\times($size of $\mathsf{G}-$order of
$\mathsf{G})$ for $\delta=4$ by adding the edges between non-adjacent vertices
of $\mathsf{G}$.

######  Theorem 4.9.

Let $G$ be a labeled connected graph of order $n\geq 2$ and $H_{i}$ be
$r_{i}$-regular graph of order $p_{i}$, where $r_{i}\geq 2$, for
$i=1,2,\ldots,n$. If $\mathsf{G}=G[H_{1},H_{2}$,…, $H_{n}]$, then the graphs
$\mathcal{L}^{k}(\mathsf{G}-v)$ satisfy the property $\rho$ for $k\geq 1$.
Moreover, if $n_{i}\geq min\\{2r_{i}:r_{i}\geq s,s\in\\{2,3,\ldots\\}\\}$,
then the graphs
$\mathcal{L}^{k}(\mathsf{G}-\\{v_{1},v_{2},\ldots,v_{2(s-1)}\\})$ satisfy the
property $\rho$ for $k\geq 1$.

###### Proof.

All the eigenvalues of $Q_{\pi}$ of $\mathsf{G}$ belongs to the union of the
closed intervals $[2r_{i},2(r_{i}+R_{i})]$ for $i=1,2,\ldots,n$ by Lemma 4.5
which shows that each eigenvalue of $Q_{\pi}$ is at least $4$ as $r_{i}\geq
2$. Each row of the matrix $Q_{\pi}$ has at least one non-diagonal entry at
least $3$ as $r_{i}\geq 2$ and $G$ is connected of order $n\geq 2$. By
Proposition 3.2 the $q_{min}$ of $\mathsf{G}$ is at least $3$. And by Theorem
2.9 the $q_{min}$ of $\mathsf{G}-v$ is at least $2$. Hence,
$\mathcal{L}(\mathsf{G}-v)$ satisfies the property $\rho$ by relation (1). If
$n_{i}\geq min\\{2r_{i}:r_{i}\geq s,s\in\\{2,3,\ldots\\}\\}$, then the
$q_{min}$ of $Q_{\pi}$ is at least $2s$, and each row of $Q_{\pi}$ has at
least one non-diagonal entry at least $2s$ with $s\in\\{2,3,\ldots\\}$. By
Proposition 3.2 the $q_{min}$ of $\mathsf{G}$ is at least $2s$. It is easy to
observe by Theorem 2.9 that $q_{min}(\mathsf{G})-2(s-1)\leq
q_{min}\big{(}\mathsf{G}-\\{v_{1},v_{2},\ldots,v_{2(s-1)}\\}\big{)}$. Hence,
$\mathcal{L}\big{(}\mathsf{G}-\\{v_{1},v_{2},\ldots,v_{2(s-1)}\\}\big{)}$
satisfies the property $\rho$ by relation (1). The graphs $\mathsf{G}-v$ for
$r_{i}\geq 2$ and $\mathsf{G}-\\{v_{1},v_{2},\ldots,v_{2(s-1)}\\}$ for
$n_{i}\geq min\\{2r_{i}:r_{i}\geq s,s\in\\{2,3,\ldots\\}\\}$ both are graphs
with minimum degree $\delta\geq 3$. Thus by Corollary 4.3 all the iterated
line graphs $\mathcal{L}^{k}(\mathsf{G}-v)$ and
$\mathcal{L}^{k}(\mathsf{G}-\\{v_{1},v_{2},\ldots,v_{2(s-1)}\\})$ for $k\geq
1$ satisfy the property $\rho$. ∎

The following is an interesting result due to the minimum number of edges in
join of two connected graphs.

######  Corollary 4.10.

If $m,n\geq 3$, then the graphs
$\mathcal{L}^{k}\big{(}K_{2}[P_{n},P_{m}]\big{)}$ satisfy the property $\rho$
for $k\geq 1$.

###### Proof.

Let $G=K_{2},H_{1}=C_{n+1}$ and $H_{2}=C_{m+1}$ for $m,n\geq 3$ in the above
Theorem. Let $v_{1}\in H_{1}$ and $v_{2}\in H_{2}$, then deleting the vertices
$v_{1}$ and $v_{2}$ and the edges incident to them respectively in $H_{1}$ and
$H_{2}$ we get $\mathsf{G}-\\{v_{1},v_{2}\\}=K_{2}[P_{n},P_{m}]$. Hence, the
graphs $\mathcal{L}^{k}(K_{2}[P_{n},P_{m}])$ satisfy the property $\rho$ for
$k\geq 1$. ∎

######  Definition 4.11.

[17] Let $v$ be the vertex of the complete graph $K_{n}$, $n\geq 3$ and let
$e_{i}$, $i=1,\ldots,p$, $1\leq p\leq n-1$ be its distinct edges all being
incident to $v$. The graph $Ka_{n}(p)$ is obtained by deleting $e_{i}$,
$i=1,\ldots,p$ from $K_{n}$, where $Ka_{n}(0)=K_{n}$.

With this notation we have the following result.

######  Theorem 4.12.

If $n\geq 6$, then the graphs $\mathcal{L}^{k}\big{(}Ka_{n}(p)\big{)}$, $1\leq
p\leq n-4$ satisfy the property $\rho$ for $k\geq 1$.

###### Proof.

All the graphs of order up to 5 with their line graphs satisfying the property
$\rho$ are $C_{4},K_{4},K_{3,2},K_{5}$ by Theorem 6.7. It is noted that none
of these graphs are of the kind $Ka_{n}(p)$ for $p\geq 1$. If $n\geq 6$, the
graph $Ka_{n}(n-4)=P_{3}[K_{1},K_{3},K_{n-4}]$. The quotient matrix $Q_{\pi}$
of $Ka_{n}(n-4)$ is

$\begin{bmatrix}3&\phantom{-}3&\phantom{-}0\\\
1&\phantom{-}n+1&\phantom{-}n-4\\\
0&\phantom{-}3&\phantom{-}2n-7\end{bmatrix}$

with its spectrum
$Sp(Q_{\pi})=\\{n-\frac{1}{2}+\frac{1}{2}\sqrt{4n(n-7)+73},n-2,n-\frac{1}{2}-\frac{1}{2}\sqrt{4n(n-7)+73}\\}$.
The signless Laplacian spectrum of $Ka_{n}(n-4)$ is
$Sp_{Q}\big{(}Ka_{n}(n-4)\big{)}=\\{(n-2)^{2},(n-3)^{n-5}\\}\cup Sp(Q_{\pi})$.
It is clear that all signless Laplacian eigenvalues of $Ka_{n}(n-4)$ are
greater than or equal to $2$ except the doubt about the third eigenvalue of
$Sp(Q_{\pi})$ for $n\geq 6$. But this eigenvalue
$n-\frac{1}{2}-\frac{1}{2}\sqrt{4n(n-7)+73}\geq 2$ if $n\geq 5$. Thus
$\mathcal{L}\big{(}Ka_{n}(p)\big{)}$ for $1\leq p\leq n-4$ satisfies the
property $\rho$ by using Proposition 2.8 and the relation (1). It is easy to
observe that the minimum degree of $Ka_{n}(p)$ for $1\leq p\leq n-4$ is at
least $3$. Hence, by Corollary 4.3 all the iterated line graphs
$\mathcal{L}^{k}(Ka_{n}(p))$ for $k\geq 1$ and $1\leq p\leq n-4$ satisfy the
property $\rho$. ∎

There are some class of graphs with least eigenvalue $-2$ like exceptional
graphs and generalized line graphs [9]. If minimum degree $\delta\geq 4$ on
these class of graphs, we have the following simple result.

######  Theorem 4.13.

Let $G$ be a graph with least eigenvalue $-2$ and minimum degree $\delta\geq
4$. Then the iterated line graphs $\mathcal{L}^{k}(G)$ satisfy the property
$\varrho$ for $k\geq 1$.

###### Proof.

The least eigenvalues $\lambda_{min},q_{min}$ and the minimum degree $\delta$
of $G$ by Proposition 2.7 satisfy $q_{min}\geq\lambda_{min}+\delta\geq-2+4=2$.
Thus by using the relation (1) $\mathcal{L}(G)$ satisfies the property $\rho$.
Now by using Corollary 4.3 all the iterated line graphs $\mathcal{L}^{k}(G)$
satisfy the property $\rho$ for $k\geq 1$. ∎

Theorem 2.5 of K. C. Das et al. can be generalized to the iterated line
graphs.

######  Theorem 4.14.

Let $G$ be a graph of order $n_{0}>2$ and size $m_{0}$ with minimum degree
$\delta\geq\frac{n_{0}}{2}+1$. Then the graphs $\mathcal{L}^{k}(G)$ satisfy
the property $\rho$ for $k\geq 1$.

###### Proof.

If $G$ be a graph of order $n_{0}>2$ and size $m_{0}$ with minimum degree
$\delta\geq\frac{n_{0}}{2}+1$, then by Theorem 2.5 $\mathcal{L}(G)$ satisfies
the property $\rho$. The existence of a graph $G$ with minimum degree
$\delta\geq\frac{n_{0}}{2}+1$ implies $n_{0}\geq 4$, that is the minimum
degree of $G$ is at least $3$. Hence by using Corollary 4.3 all the iterated
line graphs $\mathcal{L}^{k}(G)$ satisfy the property $\rho$ for $k\geq 1$. ∎

The following inequality was given by Leonardo de Lima et al. in [12] for
Turán graph.

$(r-2)\Big{\lfloor}\frac{n}{r}\Big{\rfloor}<q_{min}\big{(}T_{r}(n)\big{)}\leq\Big{(}1-\frac{1}{r}\Big{)}n.$

######  Remark 4.15.

The above inequality is not valid when $r=3$, $n=6,7,8$ and $r=4$, $n=5$ as
$Sp_{Q}\big{(}T_{3}(6)\big{)}=\\{8,4^{3},2^{2}\\}$,
$Sp_{Q}\big{(}T_{3}(7)\big{)}=\\{9.2745,5^{2},4^{2},3$, $1.7251\\}$,
$Sp_{Q}\big{(}T_{3}(8)\big{)}=\\{10.6056,6,5^{4},3.3944,2\\}$ and
$Sp_{Q}\big{(}T_{4}(5)\big{)}$ = $\\{7.3723,3^{3},1.6277\\}$, which shows that
$q_{min}\big{(}T_{3}(6)\big{)}=2,q_{min}\big{(}T_{3}(7)\big{)}=1.7251$,
$q_{min}\big{(}T_{3}(8)\big{)}=2$ and $q_{min}\big{(}T_{4}(5)\big{)}=1.6277$
but the inequality gives strict lower bound $2$ for
$q_{min}\big{(}T_{r}(n)\big{)}$. However with the help of this inequality we
have the following result.

######  Proposition 4.16.

If $r=3$ and $n\geq 6$, $n\neq 7$ or $r=4$ and $n\neq 5$ or $r\geq 5$, then
the graphs $\mathcal{L}^{k}\big{(}T_{r}(n)\big{)}$ satisfy the property $\rho$
for $k\geq 1$.

###### Proof.

The condition on $r$, $n$ in the hypothesis and the above inequality ensure
that $q_{min}\big{(}T_{r}(n)\big{)}$ is at least $2$. Thus,
$\mathcal{L}\big{(}T_{r}(n)\big{)}$ satisfies the property $\rho$ by using the
relation (1). It is easy to observe that the minimum degree of $T_{r}(n)$ is
at least $3$. Hence by Corollary 4.3 all the iterated line graphs
$\mathcal{L}^{k}(G)$ satisfy the property $\rho$ for $k\geq 1$. ∎

###  4.1 Iterated regular line graphs with property $\rho$

Most of the results so far discussed are non-regular iterated line graphs
graphs satisfying the property $\rho$. One can get iterated regular line
graphs $\mathcal{L}^{k}(G)$ satisfying property $\rho$ from Theorem 4.1 for
$k\geq 2$ and from Theorem 4.6, Theorem 4.13 and Theorem 4.14 for $k\geq 1$.
In the Proposition 4.16 if $r$ divides $n$, then we get iterated regular line
graphs $\mathcal{L}^{k}(G)$ satisfying property $\rho$ for $k\geq 1$. Here, we
present some more iterated regular line graphs $\mathcal{L}^{k}(G)$ for $k\geq
1$ by taking regular graph $G$.

######  Theorem 4.17.

If $G$ is a $r$-regular graph of order $n$ with $3\leq r\leq\frac{n-1}{3}$,
then the graphs $\mathcal{L}^{k}(\overline{G})$ satisfy the property $\rho$
for $k\geq 1$.

###### Proof.

Let $Sp_{A}(G)=\\{r,\lambda_{2}^{m_{2}},\ldots,\lambda_{t}^{m_{t}}\\}$ such
that $1+\sum\limits_{i=2}^{t}m_{i}=n$. Then by Theorem 2.3, $\overline{G}$ is
also a regular graph with
$Sp_{A}(\overline{G})=\\{n-r-1,(-1-\lambda_{t})^{m_{t}},\ldots,(-1-\lambda_{2})^{m_{2}}\\}$
and by Theorem 2.4,
$Sp_{A}(\mathcal{L}(\overline{G}))=\\{2(n-1)-2r-2,(n-r-\lambda_{t}-4)^{m_{t}},\ldots,(n-r-\lambda_{2}-4)^{m_{2}},-2^{n(n-r-3)/2}\\}$.
We shall prove that all eigenvalues of $\mathcal{L}(\overline{G})$ except $-2$
are non-negative. Since, $G$ is regular $\mathcal{L}(\overline{G})$ is also
regular with degree $2(n-1)-2r-2$ which implies $2(n-1)-2r-2$ is positive. The
condition $r\leq\frac{n-1}{3}$ gives that $n\geq 3r+1$ which implies
$n-r-\lambda_{i}-4\geq 2r-\lambda_{i}-3\geq 0$ as $r\geq 3$ to each
$i=2,\ldots,t$. Hence, $\mathcal{L}(\overline{G})$ satisfies the property
$\rho$. Now, $n\geq 3r+1$ implies $n-r-1\geq 2r\geq 6$ which shows that
$\overline{G}$ is a regular graph with its least degree$\geq 6$. This implies
the graphs $\mathcal{L}^{k}(\overline{G})$ satisfy the property $\rho$ to each
$k\geq 1$ by Corollary 4.3. ∎

######  Theorem 4.18.

If $G$ is a $r$-regular graph of order $n\geq 8$ and $r\geq 1$, then the
graphs $\mathcal{L}^{k}\big{(}\overline{\mathcal{L}(G)}\big{)}$, $k\geq 1$
satisfy the property $\rho$.

###### Proof.

Let $Sp_{A}(G)=\\{r,\lambda_{2}^{m_{2}},\ldots,\lambda_{t}^{m_{t}}\\}$ such
that $1+\sum\limits_{i=2}^{t}m_{i}=n$. Then by Theorem 2.4,
$Sp_{A}(\mathcal{L}(G))=\\{2r-2,(\lambda_{2}+r-2)^{m_{2}},\ldots,(\lambda_{t}+r-2)^{m_{t}},-2^{n(r-2)/2}\\}$
and since $\mathcal{L}(G)$ is regular by Theorem 2.3
$\overline{\mathcal{L}(G)}$ also a regular graph with
$Sp_{A}\big{(}\overline{\mathcal{L}(G)}\big{)}=\\{nr/2-2r+1,1^{n(r-2)/2},(1-r-\lambda_{t})^{m_{t}},\ldots,(1-r-\lambda_{2})^{m_{2}}\\}$.
Again, by Theorem 2.4,
$Sp_{A}\big{(}\mathcal{L}\big{(}\overline{\mathcal{L}(G)}\big{)}\big{)}=\\{r(n-4),((n-4)r/2)^{n(r-2)/2},((n-6)r/2-\lambda_{t})^{m_{t}},\ldots,((n-6)r/2-\lambda_{2})^{m_{2}},-2^{nr(nr-4r-2)/8}\\}$.
It is easy to see that all the eigenvalues of
$\mathcal{L}\big{(}\overline{\mathcal{L}(G)}\big{)}$ are non-negative except
$-2$ for $n\geq 8$. Hence $\mathcal{L}\big{(}\overline{\mathcal{L}(G)}\big{)}$
satisfies the property $\rho$. It is noted that the degree of
$\overline{\mathcal{L}(G)}$ is $nr/2-2r+1$ which is at least $3$ for $n\geq 8$
and $r\geq 1$. This implies the graphs
$\mathcal{L}^{k}\big{(}\overline{\mathcal{L}(G)}\big{)}$ satisfy the property
$\rho$ for $k\geq 1$ by Corollary 4.3. ∎

######  Remark 4.19.

One can easily construct equienergetic graphs like in Theorem 4.6 for the
results in Theorem 4.9 to Theorem 4.18.

######  Theorem 4.20.

Let $G$ be a graph with $d_{u}+d_{v}\geq 6$ to each edge $e=uv$ in $G$, then
the graphs $\mathcal{L}^{k}(G)$ are hyperenergetic for $k\geq 2$.

###### Proof.

Since $G$ is a graph with $d_{u}+d_{v}\geq 6$ to each edge $e=uv$,
$\mathcal{L}(G)$ is a graph with minimum degree $\delta\geq 4$. The number
edges in $\mathcal{L}(G)$ is
$m_{1}=\frac{1}{2}\sum\limits_{i=1}^{m_{0}}d_{i}\geq\frac{1}{2}(4m_{0})=2m_{0}=2n_{1}$,
which implies that the graph $\mathcal{L}^{2}(G)$ is hyperenergetic by Theorem
2.6. Note that the minimum degree increases in the line graphs
$\mathcal{L}^{k}(G)$ as $k$ increases for $k\geq 2$ and which is at least $4$.
Hence $m_{k}\geq 2n_{k}$, by Theorem 2.6, all the iterated line graphs
$\mathcal{L}^{k}(G)$ are hyperenergetic for $k\geq 2$. ∎

##  5 Spectra and Energy of complement of iterated line
graphs with the property $\rho$

######  Lemma 5.1.

Let $\mathcal{L}(G)$ be the line graph of a graph $G$ of order $n_{0}$ and
size $m_{0}$. If $\mathcal{L}(G)$ has non-negative eigenvalue
$\lambda_{1(j)}$, then its complement $\overline{\mathcal{L}(G)}$ has negative
eigenvalue $\overline{\lambda}_{1(m_{0}-j+2)}$ for
$j\in\\{2,3,\ldots,m_{0}\\}$. In a particular case, if $\mathcal{L}(G)$ has
eigenvalue $-2$, then its complement $\overline{\mathcal{L}(G)}$ has
eigenvalue $1$.

###### Proof.

If $G$ is a graph of order $n_{0}$ and size $m_{0}$, then its line graph
$\mathcal{L}(G)$ is the graph of order $m_{0}$. If $\mathcal{L}(G)$ has non-
negative eigenvalue $\lambda_{1(j)}$, then its complement
$\overline{\mathcal{L}(G)}$ has eigenvalue
$\overline{\lambda}_{1(m_{0}-j+2)}\leq-1-\lambda_{1(j)}$ by inequality 2,
which is negative for $j\in\\{2,3,\ldots,m_{0}\\}$. If $-2$ is the eigenvalue
of $\mathcal{L}(G)$, then by Proposition 2.11 its eigenspace is orthogonal to
all one’s vector ${\bf j}$. Now by using the Theorem 2.10, the eigenvalue
$-1-(-2)=1$ is the eigenvalue of $\overline{\mathcal{L}(G)}$. This completes
the proof. ∎

######  Theorem 5.2.

Let $G$ be a graph of order $n_{0}$ and size $m_{0}$. If the graphs
${\mathcal{L}^{k}(G)}$ satisfy the property $\rho$ with $-2$ multiplicity
$m_{k-1}-n_{k-1}$ for $k\geq 1$, then the graphs
$\overline{\mathcal{L}^{k}(G)}$ for $k\geq 1$ have exactly two positive
eigenvalues which are spectral radius and $1$ with multiplicity
$m_{k-1}-n_{k-1}$ and

$\mathcal{E}(\overline{\mathcal{L}^{k}(G)})=2\overline{\lambda}_{k(1)}+\frac{\mathcal{E}(\mathcal{L}^{k}(G))}{2}.$
(3)

###### Proof.

It is given that the iterated line graphs $\mathcal{L}^{k}(G)$ of $G$ for
$k\geq 1$ have all negative eigenvalues equal to $-2$ with the multiplicity
$m_{k-1}-n_{k-1}=n_{k}-n_{k-1}$. This implies the remaining $n_{k-1}$
eigenvalues of $\mathcal{L}^{k}(G)$ are non-negative. Now by Lemma 5.1, the
complement of iterated line graphs $\overline{\mathcal{L}^{k}(G)}$ have
negative eigenvalues $\overline{\lambda}_{k(n_{k}-j+2)}$ for
$j\in\\{2,3,\ldots,n_{k-1}\\}$ and positive eigenvalues
$\overline{\lambda}_{k(n_{k}-j+2)}=1$ for $j\in\\{n_{k-1}+1,\ldots,n_{k}\\}$.
The only remaining eigenvalue of $\overline{\mathcal{L}^{k}(G)}$ is spectral
radius $\overline{\lambda}_{k(1)}$ which must be greater than or equal to $1$
and if $\overline{\mathcal{L}^{k}(G)}$ is connected, then
$\overline{\lambda}_{k(1)}>1$. Thus $\overline{\mathcal{L}^{k}(G)}$ have
exactly two positive eigenvalues $\overline{\lambda}_{k(1)}$ and $1$ with
multiplicity $n_{k}-n_{k-1}$ for $k\geq 1$. Hence energy of
$\overline{\mathcal{L}^{k}(G)}$ is
$\mathcal{E}(\overline{\mathcal{L}^{k}(G)})=2(\overline{\lambda}_{k(1)}+n_{k}-n_{k-1})$.
But $\mathcal{E}({\mathcal{L}^{k}(G)})=4(n_{k}-n_{k-1})$ as $-2$ is only the
negative eigenvalue of ${\mathcal{L}^{k}(G)}$ with the multiplicity
$n_{k}-n_{k-1}$. Now with this we get the required energy relation between
$\mathcal{E}(\overline{\mathcal{L}^{k}(G)})$ and
$\mathcal{E}({\mathcal{L}^{k}(G)})$ for $k\geq 1$. ∎

######  Corollary 5.3.

Let $G$ be a graph of order $n_{0}$ and size $m_{0}$. If the graphs
${\mathcal{L}^{k}(G)}$ satisfy the property $\rho$ with $-2$ multiplicity
$m_{k-1}-n_{k-1}$ for $k\geq 1$, then the graphs ${\mathcal{L}^{k}(G)}$ and
$\overline{\mathcal{L}^{k}(G)}$ for $k\geq 1$ are equienergetic if and only if
$n^{-}(\mathcal{L}^{k}(G))=\overline{\lambda}_{k(1)}$.

###### Proof.

The energy relation (3) between $\mathcal{E}(\overline{\mathcal{L}^{k}(G)})$
and $\mathcal{E}({\mathcal{L}^{k}(G)})$ by Theorem 5.2 can be written as
$2\mathcal{E}(\overline{\mathcal{L}^{k}(G)})-\mathcal{E}({\mathcal{L}^{k}(G)})=4\overline{\lambda}_{k(1)}$
or
$\mathcal{E}(\overline{\mathcal{L}^{k}(G)})-\mathcal{E}({\mathcal{L}^{k}(G)})=4\overline{\lambda}_{k(1)}-\mathcal{E}(\overline{\mathcal{L}^{k}(G)})$.
Now the graphs ${\mathcal{L}^{k}(G)}$ and $\overline{\mathcal{L}^{k}(G)}$ for
$k\geq 1$ are equienergetic if and only if
$4\overline{\lambda}_{k(1)}-\mathcal{E}(\overline{\mathcal{L}^{k}(G)})=0$ or
$4\overline{\lambda}_{k(1)}=\mathcal{E}(\overline{\mathcal{L}^{k}(G)})$. But
$\mathcal{E}(\overline{\mathcal{L}^{k}(G)})=2(\overline{\lambda}_{k(1)}+n_{k}-n_{k-1})$,
now with this we get $\overline{\lambda}_{k(1)}=n_{k}-n_{k-1}$. It is given
that $n^{-}(\mathcal{L}^{k}(G))=n_{k}-n_{k-1}$ which completes the proof. ∎

######  Remark 5.4.

Recently Mojallal and Hansen in [27], the necessary and sufficient condition
for a regular graph $G$ to be equienergetic with its complement $\overline{G}$
is given. In Corollary 5.3, we extended the same to the non-regular iterated
line graphs $\mathcal{L}^{k}(G)$.

######  Corollary 5.5.

Let $G$ be a graph of order $n_{0}$ and size $m_{0}$. If the graphs
${\mathcal{L}^{k}(G)}$ satisfy the property $\rho$ with $-2$ multiplicity
$m_{k-1}-n_{k-1}$ for $k\geq 1$, then the complement of iterated line graphs
$\overline{\mathcal{L}^{k}(G)}$ of such graphs $G$ are mutually equienergetic
for $k\geq 1$ if and only if they have same spectral radius.

###### Proof.

If $G$ is the graph of order $n_{0}$ and size $m_{0}$, then the graphs
${\mathcal{L}^{k}(G)}$ have the order $n_{k}$ and size $m_{k}$. If the graphs
${\mathcal{L}^{k}(G)}$ satisfy the property $\rho$ with $-2$ multiplicity
$m_{k-1}-n_{k-1}$ for $k\geq 1$, then all the iterated line graphs
$\mathcal{L}^{k}(G)$ of such graphs $G$ of same order $n_{0}$ and same size
$m_{0}$ are mutually equienergetic with energy $4(m_{k-1}-n_{k-1})$. Now using
this fact in the energy relation (3) between
$\mathcal{E}(\overline{\mathcal{L}^{k}(G)})$ and
$\mathcal{E}({\mathcal{L}^{k}(G)})$ by Theorem 5.2 completes the proof. ∎

######  Example 5.6.

If $k\geq 1$, the graphs $\overline{\mathcal{L}^{k}(\mathsf{G_{1}})}$ and
$\overline{\mathcal{L}^{k}(\mathsf{G_{2}})}$ in the Remark 4.7 are
equienergetic as they have same quotient matrices and the fact that the
spectral radius of a quotient matrix coincides with the spectral radius of
corresponding graph. Moreover, in the Remark 4.7 if $H_{s_{1}}$ and
$H_{s_{2}}$ are non co-spectral (co-spectral) graphs, then we get non co-
spectral (co-spectral) graphs $\overline{\mathcal{L}^{k}(\mathsf{G_{1}})}$ and
$\overline{\mathcal{L}^{k}(\mathsf{G_{2}})}$ respectively for $k\geq 0$ by
using the Proposition 2.16.

######  Remark 5.7.

If $k\geq 1$ then the results in Theorem 5.2, Corollary 5.3 and Corollary 5.5
holds true for the iterated line graphs ${\mathcal{L}^{k}(G)}$ in Theorem 4.6
to Theorem 4.18. Similarly, if $k\geq 2$ these results holds true for the
iterated line graphs ${\mathcal{L}^{k}(G)}$ in Theorem 4.1 and Corollary 4.3.

######  Remark 5.8.

In [31] Ramane et al. obtained equienergetic regular graphs by means of
complement of iterated regular line graphs $\overline{\mathcal{L}^{k}(G)}$ for
$k\geq 2$ by taking regular graphs $G$ of same order and same degree $r\geq 3$
and thereby characterized large class of pairs of non-trivial equienergetic
regular graphs. It is noted that all the results in this paper become
particular case of Corollary 5.5 and Corollary 5.3.

######  Theorem 5.9.

Let $G$ be a graph with $d_{u}+d_{v}\geq 6$ to each edge $e=uv$ in $G$, then
the graphs $\overline{\mathcal{L}^{k}(G)}$ are hyperenergetic for $k\geq 2$ if
${\lambda}_{k(1)}\leq\frac{n_{k}-1}{2}$.

###### Proof.

If $G$ is the graph with $d_{u}+d_{v}\geq 6$ to each edge $e=uv$, then the
iterated line graphs $\mathcal{L}^{k}(G)$ are hyperenergetic for $k\geq 2$ by
Theorem 4.20, that is $\mathcal{E}({\mathcal{L}^{k}(G)})\geq 2(n_{k}-1)$. And
the energy relation (3) between $\mathcal{E}(\overline{\mathcal{L}^{k}(G)})$
and $\mathcal{E}({\mathcal{L}^{k}(G)})$ by Theorem 5.2 is

$\mathcal{E}(\overline{\mathcal{L}^{k}(G)})=2\overline{\lambda}_{k(1)}+\frac{\mathcal{E}(\mathcal{L}^{k}(G))}{2}\geq
2\overline{\lambda}_{k(1)}+\frac{1}{2}2(n_{k}-1)=2\overline{\lambda}_{k(1)}+(n_{k}-1).$

But it is well known that $\lambda_{k(1)}+\overline{\lambda}_{k(1)}\geq
n_{k}-1$ which implies $\overline{\lambda}_{k(1)}\geq
n_{k}-1-\lambda_{k(1)}\geq n_{k}-1-\frac{n_{k}-1}{2}=\frac{n_{k}-1}{2}$ if
${\lambda}_{k(1)}\leq\frac{n_{k}-1}{2}$. By using this we get
$\mathcal{E}(\overline{\mathcal{L}^{k}(G)})\geq
2\frac{n_{k}-1}{2}+(n_{k}-1)=2(n_{k}-1)$ which completes the proof. ∎

##  6 Other results

All the results so far discussed are non-bipartite iterated line graphs. Here
we study energy of some class of bipartite graphs by using iterated line
graphs.

######  Theorem 6.1.

Let $G$ be a graph of order $n_{0}$ and size $m_{0}$. If the graphs
${\mathcal{L}^{k}(G)}$ satisfy the property $\rho$ with $-2$ multiplicity
$m_{k-1}-n_{k-1}$ for $k\geq 1$, then
$\mathcal{E}\big{(}Ebd(\mathcal{L}^{k}(G))\big{)}=2(3m_{k-1}-2n_{k-1})$. Thus,
the graphs $Ebd(\mathcal{L}^{k}(G))$ of such graphs $G$ are mutually
equienergetic for $k\geq 1$.

###### Proof.

If the graphs ${\mathcal{L}^{k}(G)}$ satisfy the property $\rho$ with $-2$
multiplicity $m_{k-1}-n_{k-1}$ for $k\geq 1$, then the graphs
${\mathcal{L}^{k}(G)}$ have non-negative eigenvalues $\lambda_{k(j)}$ for
$j\in\\{1,\dots,n_{k-1}\\}$. Now using Theorem 2.14, the graphs
$Ebd(\mathcal{L}^{k}(G))$ have positive eigenvalues $\lambda_{k(j)}+1$ for
$j\in\\{1,\dots,n_{k-1}\\}$ and $1$ with multiplicity $m_{k-1}-n_{k-1}$. Hence
the energy of $Ebd(\mathcal{L}^{k}(G))$ is
$\mathcal{E}\big{(}Ebd(\mathcal{L}^{k}(G))\big{)}=2\big{(}\sum\limits_{j=1}^{n_{k-1}}(\lambda_{k(j)}+1)+m_{k-1}-n_{k-1}\big{)}=2\big{(}\sum\limits_{j=1}^{n_{k-1}}\lambda_{k(j)}+m_{k-1}\big{)}$.
But $\sum\limits_{j=1}^{n_{k-1}}\lambda_{k(j)}-2(m_{k-1}-n_{k-1})=0$ for the
graphs ${\mathcal{L}^{k}(G)}$, which implies
$\mathcal{E}\big{(}Ebd(\mathcal{L}^{k}(G))\big{)}=2(3m_{k-1}-2n_{k-1})$. If
there are graphs $G$ of same order $n_{0}$ and same size $m_{0}$, then all the
graphs $Ebd(\mathcal{L}^{k}(G))$ of such graphs $G$ are mutually equienergetic
with energy $2(3m_{k-1}-2n_{k-1})$. ∎

######  Theorem 6.2.

Let $G$ be a graph of order $n_{0}$ and size $m_{0}$. If the graphs
${\mathcal{L}^{k}(G)}$ satisfy the property $\rho$ with $-2$ multiplicity
$m_{k-1}-n_{k-1}$ for $k\geq 1$, then
$\mathcal{E}\big{(}Ebd(\overline{\mathcal{L}^{k}(G)})\big{)}=2\big{(}2(\overline{\lambda}_{k(1)}+1)+3m_{k-1}-4n_{k-1}\big{)}$.
Thus, all the graphs $Ebd\big{(}\overline{\mathcal{L}^{k}(G)}\big{)}$ of such
graphs $G$ are mutually equienergetic for $k\geq 1$ if and only if the graphs
$\overline{\mathcal{L}^{k}(G)}$ have same spectral radius.

###### Proof.

If the graphs ${\mathcal{L}^{k}(G)}$ satisfy the property $\rho$ with $-2$
multiplicity $m_{k-1}-n_{k-1}$ for $k\geq 1$, then the graphs
$\overline{\mathcal{L}^{k}(G)}$ have negative eigenvalues
$\overline{\lambda}_{k(n_{k}-j+2)}\leq-1$ for $j\in\\{2,3,\ldots,n_{k-1}\\}$
and two positive eigenvalues, spectral radius $\overline{\lambda}_{k(1)}$ and
$1$ with multiplicity $m_{k-1}-n_{k-1}$ by Theorem 5.2. Now using Theorem
2.14, the graphs $Ebd\big{(}\overline{\mathcal{L}^{k}(G)}\big{)}$ have non-
negative eigenvalues $\overline{\lambda}_{k(1)}+1$, $2$ with multiplicity
$m_{k-1}-n_{k-1}$ and $-(\overline{\lambda}_{k(n_{k}-j+2)}+1)$ for
$j\in\\{2,3,\ldots,n_{k-1}\\}$. Hence, the energy of
$Ebd\big{(}\overline{\mathcal{L}^{k}(G)}\big{)}$ is
$\mathcal{E}\big{(}Ebd\big{(}\overline{\mathcal{L}^{k}(G)}\big{)}\big{)}$ =
$2\big{(}\overline{\lambda}_{k(1)}+1+2(m_{k-1}-n_{k-1})+\sum\limits_{j=2}^{n_{k-1}}(-\overline{\lambda}_{k(n_{k}-j+2)}-1)\big{)}=2\big{(}\overline{\lambda}_{k(1)}+1+2(m_{k-1}-n_{k-1})-\sum\limits_{j=2}^{n_{k-1}}\overline{\lambda}_{k(n_{k}-j+2)}-n_{k-1}+1\big{)}$.
But
$\sum\limits_{j=2}^{n_{k-1}}\overline{\lambda}_{k(n_{k}-j+2)}+\overline{\lambda}_{k(1)}+(m_{k-1}-n_{k-1})=0$
for the graphs $\overline{\mathcal{L}^{k}(G)}$, which implies
$\mathcal{E}\big{(}Ebd\big{(}\overline{\mathcal{L}^{k}(G)}\big{)}\big{)}=2\big{(}2(\overline{\lambda}_{k(1)}+1)+3m_{k-1}-4n_{k-1}\big{)}$.
If there are graphs $G$ of same order $n_{0}$ and same size $m_{0}$, then all
the graphs $Ebd\big{(}\overline{\mathcal{L}^{k}(G)}\big{)}$ of such graphs $G$
are mutually equienergetic with energy
$2\big{(}2(\overline{\lambda}_{k(1)}+1)+3m_{k-1}-4n_{k-1}\big{)}$ for $k\geq
1$ if and only if $\overline{\lambda}_{k(1)}$ is same for all the graphs
$Ebd\big{(}\overline{\mathcal{L}^{k}(G)}\big{)}$ as $\mathcal{L}^{k}(G)$ have
same order $n_{k}$ and same size $m_{k}$. ∎

######  Remark 6.3.

Yaoping Hou and Lixin Xu in [25] studied spectra and energy of
$Ebd\big{(}{\mathcal{L}^{2}(G)}\big{)}$ and
$Ebd\big{(}\overline{\mathcal{L}^{2}(G)}\big{)}$, where $G$ is a $r$-regular
graph of degree $r\geq 3$, thereby characterized large class of pairs of non-
trivial bipartite equienergetic graphs $Ebd\big{(}{\mathcal{L}^{k}(G)}\big{)}$
and $Ebd\big{(}\overline{\mathcal{L}^{k}(G)}\big{)}$ for $k\geq 2$. It is
noted that all these results become particular case of Theorem 6.1 and Theorem
6.2.

######  Example 6.4.

The graphs $Ebd\big{(}\overline{\mathcal{L}^{k}(\mathsf{G_{1}})}\big{)}$ and
$Ebd\big{(}\overline{\mathcal{L}^{k}(\mathsf{G_{2}})}\big{)}$ in the Remark
4.7 are equienergetic for $k\geq 1$ as the graphs
$\overline{\mathcal{L}^{k}(\mathsf{G_{1}})}$ and
$\overline{\mathcal{L}^{k}(\mathsf{G_{2}})}$ have same quotient matrices and
the fact that the spectral radius of a quotient matrix coincides with the
spectral radius of corresponding graph. Moreover, in the Remark 4.7 if
$H_{s_{1}}$ and $H_{s_{2}}$ are non co-spectral (co-spectral) graphs, then we
get non co-spectral (co-spectral) graphs
$Ebd\big{(}\overline{\mathcal{L}^{k}(\mathsf{G_{1}})}\big{)}$ and
$Ebd\big{(}\overline{\mathcal{L}^{k}(\mathsf{G_{2}})}\big{)}$ respectively for
$k\geq 0$ by using Proposition 2.16.

######  Theorem 6.5.

Let $G$ be a graph of order $n_{0}$ and size $m_{0}$. If the graphs
${\mathcal{L}^{k}(G)}$ satisfy the property $\rho$ with $-2$ multiplicity
$m_{k-1}-n_{k-1}$ for $k\geq 1$, then the graphs
$Ebd\big{(}{\mathcal{L}^{k}(G)}\big{)}$ and
$Ebd\big{(}\overline{\mathcal{L}^{k}(G)}\big{)}$ are equienergetic for $k\geq
1$ if and only if
$\overline{\lambda}_{k(1)}=n^{-}(\overline{\mathcal{L}^{k}(G)})$.

###### Proof.

If the graphs ${\mathcal{L}^{k}(G)}$ satisfy the property $\rho$ with $-2$
multiplicity $m_{k-1}-n_{k-1}$ for $k\geq 1$, then by Theorem 6.1, we have
$\mathcal{E}\big{(}Ebd(\mathcal{L}^{k}(G))\big{)}=2(3m_{k-1}-2n_{k-1})$ and
$\mathcal{E}\big{(}Ebd(\overline{\mathcal{L}^{k}(G)})\big{)}=2\big{(}2(\overline{\lambda}_{k(1)}+1)+3m_{k-1}-4n_{k-1}\big{)}$
from Theorem 6.2. With these, we get
$\mathcal{E}\big{(}Ebd(\overline{\mathcal{L}^{k}(G)})\big{)}=\mathcal{E}\big{(}Ebd(\mathcal{L}^{k}(G))\big{)}+2\big{(}2(\overline{\lambda}_{k(1)}+1)-2n_{k-1}\big{)}$
or
$\mathcal{E}\big{(}Ebd(\overline{\mathcal{L}^{k}(G)})\big{)}-\mathcal{E}\big{(}Ebd(\mathcal{L}^{k}(G))\big{)}=4\big{(}\overline{\lambda}_{k(1)}+1-n_{k-1}\big{)}$.
Therefore, the graphs $Ebd\big{(}{\mathcal{L}^{k}(G)}\big{)}$ and
$Ebd\big{(}\overline{\mathcal{L}^{k}(G)}\big{)}$ are equienergetic for $k\geq
1$ if and only if $\overline{\lambda}_{k(1)}=n_{k-1}-1$. But
$n_{k-1}-1=n^{-}(\overline{\mathcal{L}^{k}(G)})$ by Theorem 5.2, which
completes the proof. ∎

######  Proposition 6.6.

Let $G$ be a graph of order $n_{0}$ and size $m_{0}$. If the graphs
${\mathcal{L}^{k}(G)}$ satisfy the property $\rho$ with $-2$ multiplicity
$m_{k-1}-n_{k-1}$ for $k\geq 1$, then
$\alpha\big{(}\mathcal{L}^{k}(G)\big{)}\leq n_{k-1}$ and
$\alpha\big{(}\overline{\mathcal{L}^{k}(G)}\big{)}\leq
n^{-}\big{(}\mathcal{L}^{k}(G)\big{)}+1$.

###### Proof.

If $G$ is a graph of order $n_{0}$, it is well known that $n^{-}(G)\leq
n_{0}-\alpha(G)$. Using this fact for the graphs ${\mathcal{L}^{k}(G)}$, we
get $m_{k-1}-n_{k-1}=n_{k}-n_{k-1}\leq
n_{k}-\alpha\big{(}\mathcal{L}^{k}(G)\big{)}$ which implies
$\alpha\big{(}\mathcal{L}^{k}(G)\big{)}\leq n_{k-1}$ for $k\geq 1$. Again
using the fact for the graphs $\overline{\mathcal{L}^{k}(G)}$, we get
$n_{k-1}-1\leq n_{k}-\alpha\big{(}\overline{\mathcal{L}^{k}(G)}\big{)}$ or
$-(n_{k}-n_{k-1}+1)\leq-\alpha\big{(}\overline{\mathcal{L}^{k}(G)}\big{)}$
which gives $\alpha\big{(}\overline{\mathcal{L}^{k}(G)}\big{)}\leq
n^{-}\big{(}\mathcal{L}^{k}(G)\big{)}+1$ for $k\geq 1$. ∎

The following is an interesting result on minimum order of a graph when the
graph $G$ and its complement both connected with $\mathcal{L}(G)$ satisfying
the property $\rho$ is considered.

######  Theorem 6.7.

The least order of a connected graph $G$ with its line graph satisfying the
property $\varrho$ and connected $\overline{G}$ is $7$.

###### Proof.

There are exactly $13$ non-isomorphic connected graphs of order up to 6 with
their line graphs satisfying the property $\varrho$. They are
$C_{4},K_{4},K_{3,2},K_{5},K_{4,2},K_{3,3},K_{6}$ and the following graphs in
Figure 2.

One can see that none of the above graphs have connected complement. The only
connected graph of order $7$ with its line graph satisfying the property
$\varrho$ and having connected complement is given in Figure 3, which
completes the proof.

∎

## Conclusion

Our study on eigenvalues of iterated line graphs reveals that the majority of
these graphs have exact number of negative eigenvalues which are all equal to
$-2$. Also, complements these graphs have exact number of positive eigenvalues
one more than that of multiplicity of $-2$. Although we have presented many
classes of iterated line graphs which have all negative eigenvalues equal to
$-2$ but still the question remains to characterize all such graphs.

## Acknowledgment

The author B. Parvathalu was partially funded by Karnatak University, Dharwad
under the Research Seed Grant No. KU/PMEB/2021-22/183/724.

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# Study of COVID-19 epidemiological evolution in India with a multi-wave SIR
model

Kalpita Ghosh<EMAIL_ADDRESS>Charuchandra College, 22 Lake Place
Road, Kolkata 700029, India  Asim Kumar Ghosh
<EMAIL_ADDRESS>Department of Physics, Jadavpur University,
188 Raja Subodh Chandra Mallik Road, Kolkata 700032, India

###### Abstract

The global pandemic due to the outbreak of COVID-19 ravages the whole world
for more than two years in which all the countries are suffering a lot since
December 2019. In order to control this ongoing waves of epidemiological
infections, attempts have been made to understand the dynamics of this
pandemic in deterministic approach with the help of several mathematical
models. In this article characteristics of a multi-wave SIR model have been
studied which successfully explains the features of this pandemic waves in
India. Stability of this model has been studied by identifying the equilibrium
points as well as by finding the eigen values of the corresponding Jacobian
matrices. Complex eigen values are found which ultimately give rise to the
oscillatory solutions for the three categories of populations, say,
susceptible, infected and removed. In this model, a finite probability of the
recovered people for becoming susceptible again is introduced which eventually
lead to the oscillatory solution in other words. The set of differential
equations has been solved numerically in order to obtain the variation for
numbers of susceptible, infected and removed people with time. In this
phenomenological study, finally an additional modification is made in order to
explain the aperiodic oscillation which is found necessary to capture the
feature of epidemiological waves particularly in India.

##  I INTRODUCTION

The outbreak of COVID-19 in December 2019, due to the spreading of infectious
corona virus named SARS-CoV-2 finally triggered to a global pandemic in which
all the countries are suffering a lot till to date. This highly contagious
disease has been transmitted to millions of people globally where a fraction
of infected people is succumbed to it eventually. Most of the countries
witness multiple peaks of epidemiological infections in its evolution which is
counted as number of waves. Among the most affected countries, for example,
countries in European union (CEU), USA, Russia and Canada experience five
successive epidemiological waves while the third wave is going on in India,
Indonesia and Brazil. Which means that five peaks of the epidemiological
infections are found in CEU, USA, Russia and Canada, while three distinct
peaks are noted in India, Indonesia and Brazil. However, peaks of
epidemiological infections in India are sharper than those found in Brazil.
So, it is obvious that characteristics of these epidemiological waves found in
different countries are not similar. Widths and heights of the peaks are
different as well as the separation between them are not equal. It is observed
that the same individual gets infected multiple time during this pandemic in
all countries.

In order to study the dynamics of the pandemic in a deterministic approach,
several models have been proposed based on the classic SIR model introduced by
Kermack and Mckendrick in 1927. SIR  Most of the models are mainly formulated
to study the characteristics of the first wave of the pandemic. Those are
known as SEIR, SIQR and their hybrids which are all derived from the single-
wave SIR (SWSIR) model. Hethcote ; Dickmann-Hethcote ; Keeling-Rohani ;
Martcheva ; Daley-Gani  It is termed as SWSIR model because of the fact that
this primitive version in general yields a solitary peak in its evolution. In
those attempts, effects of quarantine, isolation, latent time of infection and
other factors are taken into account where several predictions and forecasts
are available. Hethcote-Zhien-Shengbing ; Jumpen-Wiwatanapataphee-Wu-Tang
Nonetheless, periodic outbreak of disease in terms of SIR, SIQR, and SEIQR
models are studied analytically in great details. Hethcote-Levin ; Feng-
Thieme1 ; Feng-Thieme2 ; Feng-Thieme3  Evolution of the COVID-19 pandemic for
different countries have been studied with the help of those models where
origin of differences in its features are explained. Case studies on few
countries are available. China ; Italy ; NewZealand ; Brazil ; France ; UK-
Israel ; Canada

Hence, a multi-wave SIR (MWSIR) model is formulated by modifying the SWSIR
model in order to explain the dynamics of epidemiological infections including
the origin of multiple waves found in the infection pattern. This is a fact
that the same individual has been infected repeatedly during this pandemic
period. So in this model, probability of the same individual for getting
infected multiple times after curing is taken into account.SODE1  Time delay
between successive infections of the same individual is introduced in the
MWSIR model. In this work, periodic outbreak of COVID-19 in India will be
studied, and so, the daily new cases in terms of seven-day rolling average in
India during January 30, 2020 to February 3, 2022 (736 days) are shown in
Figure 1.

Figure 1: Daily new COVID-19 cases in terms of seven-day rolling average in
India during January 30, 2020 to February 3, 2022 (736 days). The regions for
first, second and ongoing third waves are demarcated by different colors.

It reveals that the duration of those waves are different, so, it is not
perfectly periodic. The approximate duration of first and second waves are
$T_{1}=370$ and $T_{2}=300$ days, respectively, while the third wave crosses
66 days ($T_{3}>66$) by this time. Hence, separation between adjacent peaks
are different. In addition to that width and height of the peaks are unequal.

In the section II a MWSIR model is considered which gives rise to almost
periodic epidemiological oscillation. The stability of this model is studied
in section III analytically while its periodic feature is studied numerically
in section IV. In order to produce aperiodic oscillation, this model has been
further modified. The epidemiological oscillation produced by this model is
compared with the daily new COVID-19 cases in India. A discussion based on
those results is available in section V.

##  II Multi-wave SIR model

In order to explain the epidemiological waves of infection due to the
spreading of COVID-19 in India, a multi-wave SIR model is formulated where the
total population is divided into three dynamic sub-populations, those are
known as susceptible, infected and removed. Susceptible population is
constituted by the individuals those are otherwise healthy but have a
probability for having infected at any time in future. The intermediate stage
is comprised of infected population where the individual is instantaneously
contagious. In the terminal stage individuals are either recovered or
succumbed to the disease, however, jointly referred to as removed population.
As the ongoing COVID-19 pandemic exhibits periodic outbreak of the disease
globally, a new model is necessary to study the dynamics of this pandemic.

Therefore, in order to understand the nature of periodic outbreak, a MWSIR
model has been formulated by modifying the primitive version of SIR model as
described below. In this model the recovered population has a nonzero
probability to get infected again. As a result, the susceptible population
gets enriched with time which is measured in terms of infected population with
a time delay, $\tau$. Thus, the multi-wave SIR model is defined by a set of
coupled first-order nonlinear differential equations (Eq 1):

$\left\\{\begin{array}[]{l}\\!\frac{d{S}}{dt}=\mu
N-\alpha\,{S(t)}\,{I(t)}+\gamma\,I(t\\!-\\!\tau)-\mu S(t),\\\\[5.0pt]
\\!\frac{d{I}}{dt}=\alpha\,{S(t)}\,{I(t)}-\beta\,{I(t)}-\mu I(t),\\\\[5.0pt]
\\!\frac{d{R}}{dt}=\beta\,{I(t)}-\gamma\,I(t\\!-\\!\tau)-\mu
R(t),\end{array}\right.$ (1)

where $S(t)$, $I(t)$, and $R(t)$ are the number of susceptible, infected and
removed people at time $t$. The positive constants $\mu,\alpha,\beta$ and
$\gamma$ are the rates of birth per individual, infection, removed and
susceptibility, respectively. The model incorporates the fact that the
recovered people become susceptible again after the mean period $\tau$.SODE1
However, $I(t\\!-\\!\tau)=0$, when $t<\tau$. The SWSIR model will be restored
when $\mu=0$, and $\gamma=0$. The system in this formulation is closed in a
sense that the total population, $N=S(t)+I(t)+R(t)$, does not change with
time, which means the rate at which individual suffers natural mortality is
also given by the parameter $\mu$. In other words, mean lifespan of individual
turns out to be $1/\mu$. In the same way, $1/\beta$ can be regarded as the
mean recovery time for the infected individual. The flowchart of this model is
shown in Figure 2.

Figure 2: Flowchart for the MWSIR model.

The basic reproduction number, $R_{0}$ plays an important role in
epidemiology, which is defined as average number of new infections per
infected individual. So, in this three-compartment system, it is equal to the
transmission rate multiplied by the infectious period,
$R_{0}=\alpha/(\beta+\mu)$. Keeling-Rohani ; Diekmann-Heesterbeek-Roberts  The
value of $R_{0}$ is crucial to have an idea about how does the disease flow in
the whole population and at the same time it provides clue to control its
spreading.

##  III Stability analysis of multi-wave SIR model

In order to determine the stable equilibrium points, the rate of change of
$S(t)$, $I(t)$ and $R(t)$ with time are made zero. The trivial solution leads
to $I(t)=0$ at any time, which corresponds to disease-free equilibrium (DFE).
Mathematically, this point is expressed as $(S^{*},I^{*},R^{*})_{\rm
tr}=(N,0,0)$, where the entire population becomes susceptible but no infected
individual. This is true for both $t<\tau$ and $t\geq\tau$. Anyway, this DFE
is insignificant as it does not correspond to dynamics of epidemic by any
means.

The nontrivial solution leads to $I(t)\neq 0$, at finite time, but,
$I(t\rightarrow\infty)\rightarrow 0$, which corresponds to endemic equilibrium
and that is observed in SWSIR model. However, in this MWSIR model, nontrivial
solutions mean $I(t)>0$, at any time. So these dynamic equilibrium points
appear periodically with time without showing any signature of ending if there
is no damping. These equilibrium points are marked by
$(S^{*},I^{*},R^{*})_{\rm ntr}=\left(\frac{\beta+\mu}{\alpha},\frac{\mu
N}{\beta+\mu}-\frac{\mu}{\alpha},\frac{\beta
N}{\beta+\mu}-\frac{\beta+\mu}{\alpha}+\frac{\mu}{\alpha}\right)$, when
$t<\tau$, however, for $t\geq\tau$, $(S^{*},I^{*}(t),R^{*}(t))_{\rm
ntr}=\left(\frac{\beta+\mu}{\alpha},\frac{\mu N}{\beta+\mu}+\frac{\gamma
I(t\\!-\\!\tau)}{\beta+\mu}-\frac{\mu}{\alpha},\frac{\beta
N}{\beta+\mu}-\frac{\beta+\mu}{\alpha}-\frac{\gamma
I(t\\!-\\!\tau)}{\beta+\mu}+\frac{\mu}{\alpha}\right)$. Hence, equilibrium
points are static when $t<\tau$, but dynamic when $t\geq\tau$.

In order to understand the dynamics close to the equilibria, Jacobian matrix
($J$) is constructed which can be expressed at the respective equilibrium
points. Jacobian matrix has the form

$\left[\begin{array}[]{ccc}-(\alpha I^{*}+\mu)&-\alpha S^{*}&0\\\\[5.0pt]
\alpha I^{*}&\alpha S^{*}-\beta-\mu&0\\\\[5.0pt]
0&\beta&-\mu\end{array}\right].$ (2)

The eigenvalues of Jacobian matrix ($\lambda_{0},\lambda_{\pm}$) can be given
as

$\left\\{\begin{array}[]{l}\\!\lambda_{0}=-\mu,\\\\[5.0pt]
\\!\lambda_{\pm}=-\frac{\alpha(I^{*}-S^{*})+\beta+2\mu}{2}\pm\frac{\sqrt{\alpha^{2}(I^{*}-S^{*})^{2}+\beta^{2}-2\alpha\beta(S^{*}+I^{*})}}{2}.\end{array}\right.$
(3)

For the DFE, $(S^{*},I^{*},R^{*})_{\rm tr}=(N,0,0)$, the eigenvalues are

$\left\\{\begin{array}[]{l}\lambda_{0}=-\mu,\\\\[5.0pt] \lambda_{+}=\alpha
N-\beta-\mu,\\\\[5.0pt] \lambda_{-}=-\mu.\end{array}\right.$ (4)

As a result, it will behave like a stable point when $\alpha<(\beta+\mu)/N$,
which is similar to the SWSIR model. It is obvious that the eigenvalues will
be complex for the general case when

$\alpha\beta>\frac{\alpha^{2}(I^{*}-S^{*})^{2}+\beta^{2}}{2(S^{*}+I^{*})}.$

It is expected that variations of $S(t)$, $I(t)$, and $R(t)$ will be
oscillatory around these points.

##  IV Numerical results

As the values of $S(t)$, $I(t)$, and $R(t)$ cannot be determined analytically
by solving the set of equations, Eq 1, they have been solved numerically. The
oscillations of $S(t)$, $I(t)$, and $R(t)$ are noted for long time and it is
observed that all are oscillating with almost constant amplitude. Four
successive waves are shown in Fig 3, where it is observed that the mean time
period of oscillations, $T$, is always greater than $\tau$. As no damping
factor is introduced here, amplitudes of the respective waves for $S(t)$,
$I(t)$, and $R(t)$ do not change with time.

The time period of oscillations for $S(t)$, $I(t)$, and $R(t)$ are obviously
the same, although it decreases with the number of cycles. However the
difference, $T\\!-\\!\tau$ vanishes with the increase of $\tau$. This feature
is shown in Fig 4 where variations of time period $T$ along with relative
difference between $T$ and $\tau$ or $(T\\!-\\!\tau)/\tau$ with $\tau$ are
plotted.

Figure 3: Four successive waves are shown here those are generated numerically
by solving MWSIR model (Eq 1). Values of the parameters:
$S(0)=10,\,I(0)=1,\,R(0)=0,\,N=11,\,\alpha=0.5,\,\beta=1,\,\gamma=1,\,\tau=10,\,\mu=0.$
Time period of oscillation is marked by $T$. Figure 4: Variation of $T$ and
$(T\\!-\\!\tau)/\tau$ with $\tau$, in the MWSIR model for
$S(0)=10,\,I(0)=1,\,R(0)=0,\,N=11,\,\alpha=0.5,\,\beta=1,\,\gamma=1,\,\mu=0.$
Figure 5: Variation of $T$ with $\alpha$ and $\beta$, when $\tau=10$, in the
MWSIR model for $S(0)=10,\,I(0)=1,\,R(0)=0,\,N=11,\,\gamma=1,\,\mu=0.$

The variation of $T$ with $\alpha$ and $\beta$ for fixed value of $\tau$ is
shown in Figure 5. It clearly indicates that $T>\tau$, for any values of
$\alpha$ and $\beta$. Moreover, the difference, $T\\!-\\!\tau$ steadily
decreases with $\alpha$ but increases gradually with $\beta$.

Figure 6: Daily new COVID-19 cases in terms of seven-day rolling average in
India during January 30, 2020 to February 3, 2022 (736 days) is compared with
the numerical results obtained by the MWSIR model. Values of the parameters
are $\alpha=1.78\times 10^{-7},\,\beta=1/14,\,S(0)=7.3\times
10^{5},\,\,I(0)=1,\,R(0)=0.$

Features of the ongoing COVID-19 epidemiological waves in India are shown in
Figure 1. It reveals that the duration of successive waves are decreasing as
$T_{1}>T_{2}>T_{3}$. On the other hand, amplitude of the second wave is the
maximum while the third wave is expected to cross its peak by this time. These
features suggest that the prevailing waves are not periodic by any means.
Hence the MWSIR model is further modified in order to explain this aperiodic
behavior. One can remember that transformation of SWSIR to MWSIR is
accomplished by introducing two parameters $\gamma$ and $\tau$, where $\gamma$
is the rate at which the recovered people become susceptible again, while
$\tau$ is the time delay to estimate the number of susceptible people in terms
of infected people.

In order to capture this specific aperiodic features in the most simple way,
the value of $\gamma$ has been split up for three different waves in the MWSIR
model as

$\gamma=\left\\{\begin{array}[]{lc}0,&t<\tau,\\\\[3.00003pt]
\gamma_{1},&\tau\leq t<\nu\,\tau,\\\\[3.00003pt]
\gamma_{2},&t\geq\nu\,\tau,\end{array}\right.$

where $\gamma_{1}=0.166$, $\gamma_{2}=0.056$, $\nu=2.78$, and $\tau=175$ days.
The relation, $\gamma_{2}<\gamma_{1}$, corresponds to the fact that height of
third peak is lower than the second one. The value, $\beta=1/14$, is kept
fixed as it corresponds to the mean recovery period of COVID-19, $1/\beta=14$
days. The numerical data obtained by solving the MWSIR model is plotted in red
dashed line and compared with the daily new COVID-19 cases (seven-day rolling
average) in India shown in blue sold line. A very good agreement is found when
the rate of infection is assumed very small ($\alpha=1.78\times 10^{-7}$).

This phenomenological study predicts that 16.6% of individuals infected during
January 30, 2020 to December 5, 2020 (first wave) become susceptible again for
further infection. Similarly, 5.6% of individuals infected during December 5,
2020 to May, 27, 2021 (second wave) become susceptible again for the next
infection. This reduction may account the effects of quarantine, isolation,
vaccination and other preventive measures. However, as the solutions of the
non-linear equations are highly sensitive to the initial conditions, like
$S(0),\,I(0)$, and $R(0)$, estimated values of the parameters,
$\alpha,\,\gamma,\,\nu$ and $\tau$ are likely to change dramatically with the
change of initial values. Which on the other hand will change the predictions
for obvious reason.

##  V Discussion

In order to study the behaviour of ongoing pandemic of COVID-19 in India
mathematically, a modified version of SWSIR model termed as MWSIR model is
proposed here which is found successful to reproduce the features of the
multiple epidemiological waves observed in this country. The model is deigned
in such a fashion that it is able to yield periodic as well as aperiodic
epidemiological waves with varying widths and heights by tuning its
parameters. As the SARS-CoV-2 virus responsible for the COVID-19 is highly
infectious, the spreading of this disease cannot be controlled easily by means
of simple preventive measures like quarantine, isolation, lock down and even
vaccination. As a result, in due course, a sizable fraction of recovered
people gets infected multiple times leading to multiple waves of the
epidemiological infection. In order to explain this particular feature, the
SIR model has been modified in such a manner that the recovered people has a
finite chance for becoming susceptible again after their recovery in this
modified version. Estimated values of the parameters obtained in this study
will be useful for imposing restrictions to control the pandemic. It will
become helpful even for making accurate forecast and prediction.

However, in this simple model no time delay for the incubation period of the
pathogens within the human body is considered, and the effect of quarantine,
isolation, vaccination and other macroscopic measures are not take into
account. So, it is expected that more accurate prediction can be made with the
help of this model by accommodating the effect of quarantine, isolation,
vaccination, lock-down and other factors in its modified version. Which means
that the discrepancy between theoretical estimation and daily new cases can be
removed further by formulating multi-wave SEIR, SIQR, SEIQR and other hybrid
models. Obviously, this model can be employed for examining the
characteristics of epidemiological waves found in other countries as well.

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