Inverse sum indeg energy of graphs

TL;DR

This paper introduces the inverse sum indeg (ISI) energy of graphs based on the ISI matrix, provides formulas for specific graph classes, and establishes bounds for this new graph energy measure.

Contribution

It defines the ISI energy of graphs, derives formulas for certain classes, and presents bounds, advancing spectral graph theory with a new energy concept.

Findings

Formulas for ISI energy of some graph classes

Bounds for ISI energy of graphs

Introduction of a new spectral graph invariant

Abstract

Suppose G is an n-vertex simple graph with vertex set {v1,..., vn} and d(i), i = 1,..., n, is the degree of vertex vi in G. The ISI matrix S(G) = [sij] of G is a square matrix of order n and is defined by sij = d(i)d(j)/d(i)+d(j) if the vertices vi and vj are adjacent and sij = 0 otherwise. The S-eigenvalues of G are the eigenvalues of its ISI matrix S(G). Recently the notion of inverse sum indeg (henceforth, ISI) energy of graphs is introduced and is defined as the sum of absolute values of S-eigenvalues of graph G. We give ISI energy formula of some graph classes. We also obtain some bounds for ISI energy of graphs.

Equations

$$a_{ij}=\left\{\begin{array}[]{ll}1&\mbox{if $v_{i}\sim v_{j}$,}\\ 0&\mbox{otherwise.}\end{array}\right.$$

$$\Phi (G,\lambda )$$

$$\text{ISI}(G)=\sum _{i\sim j}\frac{d_i{d}_j}{d_i+d_j}.$$

$$s_{ij}=\left\{\begin{array}[]{ll}\frac{d_{i}d_{j}}{d_{i}+d_{j}}&\mbox{if $v_{i}\sim v_{j}$,}\\ 0&\mbox{otherwise.}\end{array}\right.$$

$${\Phi }_S(G,\lambda )$$

$$E_{\mathrm{ISI}}(G)=\sum _{i=1}^n|{\tau }_i|.$$

$$Q=2\sum\limits_{1\leq i<j\leq n}\bigg{(}\frac{d_{i}d_{j}}{d_{i}+d_{j}}\bigg{)}^{2},~{}~{}~{}\Theta=\det(S(G)).$$

$$tr(S^2(G))\le tr(S^2(K_n)),$$

$$\frac{d_i{d}_j}{d_i+d_j}=\frac{1}{\frac{1}{d_i}+\frac{1}{d_j}}\le \frac{1}{\frac{1}{n-1}+\frac{1}{n-1}}=\frac{n-1}{2},$$

$$tr(S^{2}(G))=2\sum\limits_{1\leq i<j\leq n}\bigg{(}\frac{d_{i}d_{j}}{d_{i}+d_{j}}\bigg{)}^{2}\leq 2m\frac{(n-1)^{2}}{4}=\frac{m(n-1)^{2}}{2}.$$

$$tr(S^2(G))\le \frac{m(n-1{)}^2}{2}$$

$$tr(S^2(K_n))$$

$${\mathrm{spec}}_S(G)={\mathrm{spec}}_S(G_1)\cup {\mathrm{spec}}_S(G_2)\cup \cdots \cup {\mathrm{spec}}_S(G_s).$$

$$E_{\mathrm{ISI}}(G)=\sum _{i=1}^k{E}_{\mathrm{ISI}}(G_i).$$

$$\sum _{i=1}^s{\tau }_i+\sum _{i=s+1}^n{\tau }_i=\sum _{i=1}^n{\tau }_i$$

$$\sum _{i=1}^s{\tau }_i=-\sum _{i=s+1}^n{\tau }_i.$$

$$E_{\mathrm{ISI}}(G)$$

$$|{\tau }_1|=|{\tau }_2|=\cdots =|{\tau }_m|>0(m\ge 2)$$

$${\mathrm{spec}}_A(K_n)$$

$${\mathrm{spec}}_A(K_{m,n})$$

$$E(C_{n})=\left\{\begin{array}[]{ll}4~{}\cot\frac{\pi}{n}&\mbox{if $n\equiv 0(\bmod 4)$}\vspace{.2cm}\\ 4~{}\csc\frac{\pi}{n}&\mbox{if $n\equiv 2(\bmod 4)$}\vspace{.2cm}\\ 2~{}\csc\frac{\pi}{2n}&\mbox{if $n\equiv 1(\bmod 2)$.}\end{array}\right.$$

$$s_{ij}$$

$$S(K_{m,n})=\left[\begin{array}{cc}O& B\\ C& O^{\prime }\end{array}\right].$$

$$S(K_{m,n})=\frac{mn}{m+n}A(K_{m,n}).$$

$${\mathrm{spec}}_S(K_{m,n})=\left\{\frac{(mn{)}^{\frac{3}{2}}}{m+n},0^{(m+n-2)},-\frac{(mn{)}^{\frac{3}{2}}}{m+n}\right\}.$$

$$E_{\mathrm{ISI}}(K_{m,n})$$

$$s_{ij}$$

$$S(K_n)=\frac{n-1}{2}A(K_n).$$

$${\mathrm{spec}}_S(K_n)=\left\{{\frac{(1-n)}{2}}^{(n-1)},\frac{(n-1{)}^2}{2}\right\}.$$

$$E_{\mathrm{ISI}}(K_n)$$

$$\sqrt{\frac{tr(B^2)}{n(n-1)}}$$

Full text

††Email addresses: sumaira.hafeez123@gmail.com (S. Hafeez), farook.ra@gmail.com (R. Farooq).

Keywords: Energy of graphs; inverse sum indeg

AMS Classification: 05C35, 05C50

1 Introduction

A graph $G$ is a pair $G=(V(G),E(G))$, where $V(G)$ denotes the vertex set $\{v_{1},\dots,v_{n}\}$ and $E(G)$ denotes the edge set of $G$. The degree $d_{i}$ of a vertex $v_{i}$ is the number of edges incident on it. If vertices $v_{i}$ and $v_{j}$ are adjacent, we denote it by $v_{i}\sim v_{j}$. If vertices $v_{i}$ and $v_{j}$ are not adjacent, we denote it by $v_{i}\not\sim v_{j}$. An $n$-vertex path $P_{n}$, $(n\geq 1)$, is a graph with vertex set $\{v_{1},\dots v_{n}\}$ and edge set $\{v_{j}v_{j+1}|j=1,2\dots,n-1\}$. An $n$-vertex cycle C_{n}$$(n\geq 3) is a graph with vertex set $\{v_{1},\dots,v_{n}\}$ and edge set $\{v_{j}v_{j+1}|j=1,2,\dots,n-1\}\cup\{v_{n}v_{1}\}$. The star graph $S_{n}$ of order $n$ is isomorphic to $K_{1,n-1}$. By $\overline{G}$, we denote the complement of $G$.

The adjacency matrix $A(G)=[a_{ij}]_{n\times n}$ of an $n$-vertex graph $G$ is defined as

[TABLE]

The $A$-characteristic polynomial of $G$ is the polynomial

[TABLE]

where $I_{n}$ is the diagonal matrix of order $n$ with diagonal entries equal to 1. The $A$-eigenvalues of $G$ are the $A$-eigenvalues of $A(G)$. The spectrum of $G$, denoted by ${\rm spec}_{A}(G)$, is the set of $A$-eigenvalues of $G$ together with their multiplicities.

The inverse sum indeg, (henceforth ISI) index, was studied in [24]. The ISI index is defined as

[TABLE]

Zangi et al. [26] defined the ISI matrix $S(G)=[s_{ij}]_{n\times n}$ of an $n$-vertex graph $G$ as:

[TABLE]

The $S$-characteristic polynomial of $G$ is given by:

[TABLE]

where $I_{n}$ is the diagonal matrix of order $n$ with diagonal entries equal to 1. The $S$-eigenvalues of $G$ are the $S$-eigenvalues of $S(G)$. The $S$-spectrum ${\rm spec}_{S}(G)$ of $G$, is the set of $S$-eigenvalues of $G$ together with their multiplicities. Since ISI matrix of graph is symmetric and real, therefore its eigenvalues are real. If $G$ is an $n$-vertex graph with distinct $S$-eigenvalues $\tau_{1},\tau_{2},\dots,\tau_{k}$ and if their respective multiplicities are $p_{1},p_{2},\dots,p_{k}$, we write the $S$-spectrum of $G$ as $\rm spec_{\bf{S}}(G)=\{\tau_{1}^{(p_{1})},\tau_{2}^{(p_{2})},\dots,\tau_{k}^{(p_{k})}\}$.

In $1978$, the energy of a simple graph is defined by Gutman [11] as $E(G)=\sum\limits_{i=1}^{n}|\lambda_{i}|$. Many results on the graph energy can be found in literature. The concept of Randic energy is given by Bozkurt et al. [2, 3]. In 2014, Gutman et al. [12] gave some of the properties of Randić matrix and Randić energy. Sedlar et al. [23] study the properties ISI index and finds extremal values of ISI index for some classes of graphs. Pattabiraman [17] gave some extremal bounds on ISI index. In 2018, Das et al. [8] summarized different types of energies of graphs introduced by many authors. Das et al. [8] find some of the lower and upper bounds for these energies of graphs. For recent results on different types of energies of graphs, one can study [6, 9, 10, 16, 18, 19, 20, 21, 27].

Zangi et al. [26] introduce the concept of ISI energy of graphs. In this paper we obtain ISI energy formula of some well-known graphs. Upper and lower bounds are established. Finally, we give integral representation for ISI energy of graphs.

2 Inverse sum indeg energy

Let $\lambda_{1},\dots,\lambda_{n}$ be $A$-eigenvalues of an $n$-vertex graph $G$. Then Gutman [11] defined the energy of $G$ as $E(G)=\sum\limits_{i=1}^{n}|\lambda_{i}|$.

Let $\tau_{1},\dots,\tau_{n}$ be the $S$-eigenvalues of $G$. Then Zangi et al. [26] define ISI energy of $G$ as

[TABLE]

For convenience , we define some notations. We denote determinant of $S(G)$ by $\det(S(G))$. Let

[TABLE]

The trace of the matrix $S(G)=[s_{ij}]_{n\times n}$ is defined by $\sum\limits_{i=1}^{n}s_{ii}$ and is denoted by $tr(S(G))$. Zangi et al. [26] prove the following lemma.

Lemma 2.1** (Zongi et al. [26]).**

Let $G$ be an $n$-vertex graph and let $\tau_{1},\dots,\tau_{n}$ be its $S$-eigenvalues. Then

• $(1)$

$\sum\limits_{i=1}^{n}\tau_{i}=0$,

• $(2)$

$tr(S^{2}(G))=\sum\limits_{i=1}^{n}\tau_{i}^{2}=Q.$**

Theorem 2.2**.**

Let $G$ be an $n$-vertex simple and connected graph and let $m$ be the number of edges in $G$. Then

[TABLE]

where the equality holds if and only if $G\cong K_{n}$.

Proof.

First let $G\not\cong K_{n}$. Then $d_{i}\leq n-1$ for every vertex $v_{i}$ of $G$, $i=1,\dots,n$. Therefore

[TABLE]

[TABLE]

As $G\not\cong K_{n}$, it holds that $m<\frac{n(n-1)}{2}$. Consequently

[TABLE]

Now let $G\cong K_{n}$. Lemma 2.1 implies

[TABLE]

Hence $tr(S^{2}(G))\leq tr(S^{2}(K_{n}))$. ∎

Suppose $G_{1}$ and $G_{2}$ are two graphs with disjoint vertex sets. Then the graph union $G_{1}\cup G_{2}$ is a pair $G_{1}\cup G_{2}=(V(G_{1}\cup G_{2}),E(G_{1}\cup G_{2}))=(V(G_{1})\cup V(G_{2}),E(G_{1})\cup E(G_{2}))$. The degree of a vertex $v$ of $G_{1}\cup G_{2}$ is equal to the degree of the vertex $v$ in the component $G_{i}$, $i=1,2$, that contains it. A square diagonal matrix whose diagonal elements are square matrices and the non-diagonal elements are 0 is called a block diagonal matrix.

Next theorem gives the relation between ISI energy of a graph and its components.

Theorem 2.3**.**

Suppose $G_{1},G_{2},\dots,G_{s}$ are the components of a graph $G$. Then $E_{{\rm ISI}}(G)=\sum\limits_{i=1}^{s}E_{{\rm ISI}}(G_{i})$.

Proof.

Since $G_{1},G_{2},\dots,G_{s}$ are the components of $G$, we can write $G=G_{1}\cup G_{2}\cup\dots\cup G_{s}$. Then $S(G)$ is a block diagonal matrix with diagonal elements $S(G_{1}),S(G_{2}),\dots S(G_{s})$. Therefore

[TABLE]

Hence

[TABLE]

∎

Following result follows directly from ISI matrix of $\overline{K}_{n}$.

Lemma 2.4**.**

Suppose $G$ is an $n$-vertex graph. Then $E_{{\rm ISI}}(G)=0$ if and only if $G\cong\overline{K}_{n}$.

We now show that the ISI energy of a non-trivial graph, if it is an integer, must be an even positive integer.

Theorem 2.5**.**

If $G\not\cong\overline{K}_{n}$ and the ISI energy of a graph $G$ is an integer then it must be an even positive integer.

Proof.

Let $\tau_{1},\dots,\tau_{n}$ be $S$-eigenvalues of $G$ and with no loss of generality, assume that $\tau_{1},\dots,\tau_{s}$ are positive and $\tau_{s+1},\dots,\tau_{n}$ are non-negative. From Lemma 2.1, we have

[TABLE]

This gives

[TABLE]

Now

[TABLE]

Therefore ISI energy of $G$ is an even integer. ∎

The distance between $v$ and $u$ of $G$ is the length of the shortest path between them. The maximum distance between a vertex $v$ to all other vertices of $G$ is called the eccentricity of $v$. The diameter of $G$ is the maximum eccentricity of any vertex in $G$. A matrix $M$ is irreducible if the digraph associated with $M$ is strongly connected. A matrix is non-negative if its all entries are non-negative.

In the following two results, we determine some properties of the $S$-eigenvalues. The idea of proof is taken from proof of Lemma $1.1$ [7]

Lemma 2.6**.**

Let $G$ be an $n$-vertex simple and connected graph, $n\geq 2$, with non-incresing $S$-eigenvalues $\tau_{1}\geq\tau_{2}\geq\dots\geq\tau_{n}$. If $G$ has diameter atleast $3$, then $\tau_{1}>\tau_{2}>0$.

Proof.

Since the graph $G$ is connected therefore $S(G)$ is an irreducible non-negative square matrix of order $n$. By Perron-Frobenius theorem, we have $\tau_{1}>\tau_{2}$. Since $G$ has diameter at least 3, $P_{4}$ is the subgraph of $G$. Therefore we have $\tau_{2}(G)\geq\tau_{2}(P_{4})=\frac{4}{3}>0$, where $\tau_{2}(G)$ is the second largest $S$-eigenvalue of $G$ and $\tau_{2}(P_{4})$ is the second largest $S$-eigenvalue of $P_{4}$ . Hence $\tau_{1}>\tau_{2}>0$. ∎

Lemma 2.7** (Brouwer and Haemers [4]).**

Let $G$ be a connected graph with greatest eigenvalue $\lambda_{1}$. Then $-\lambda_{1}$ is an eigenvalue of $G$ if and only if $G$ is bipartite.

Theorem 2.8**.**

Suppose $G$ is an $n$-vertex graph, $n\geq 2$, with $S$-eigenvalues $\tau_{1},\dots,\tau_{n}$ and let its $A$-spectrum and $S$-spectrum are symmetric about the origin. Then $|\tau_{1}|=|\tau_{2}|=\dots=|\tau_{m}|>0$ $(m\geq 2)$ and the remaining $S$-eigenvalues are zero (if exist) if and only if $G\cong\bigcup\limits_{j=1}^{p}K_{r,s}$, where $p(r+s)=n$ and one of the $r$ or $s$ is greater than 1.

Proof.

First assume that

[TABLE]

and the remaining $S$-eigenvalues are zero (if exist). Then each component of $G$ has atmost three distinct $S$-eigenvalues. Let $H$ be a component of $G$. From (2.3) and Lemma 2.7, we see that $H$ is bipartite. If $H$ is not a complete bipartite graph, then the diameter of $H$ is at least 3. Therefore by Lemma 2.6 and (2.3), we get a contradiction. Hence $H$ is a complete bipartite graph. As $H$ is arbitrary component of $G$, therefore $G\cong\bigcup\limits_{j=1}^{p}K_{r,s}$, where $p(r+s)=n$.

The converse statement is easy to prove. ∎

3 ISI energy of some graphs

In this section, we prove ISI energy formulae for some classes of graphs.

The $A$-spectrum of $K_{n}$ and $K_{m,n}$ is given by

[TABLE]

Bhat and Pirzada [1] gave the following energy formulae for cycle $C_{n}$ of order $n$:

[TABLE]

Theorem 3.1**.**

$E_{{\rm ISI}}(C_{n})=E(C_{n})$**

Proof.

Since degree of every vertex in $C_{n}$ is $2$, therefore for any $v_{i},v_{j}\in V(C_{n})$ with $v_{i}\sim v_{j}$, we have

[TABLE]

If $v_{i}\not\sim v_{j}$ then $s_{ij}=0$. Hence $S(C_{n})=A(C_{n})$. Consequently ${\rm spec}_{S}(G)={\rm spec}_{A}(G)$ and $E_{{\rm ISI}}(C_{n})=E(C_{n})$. ∎

Theorem 3.2**.**

$E_{{\rm ISI}}(K_{m,n})=\frac{2(mn)^{\frac{3}{2}}}{m+n}$.

Proof.

Let $B$ be an $m\times n$ matrix and $C$ be an $n\times m$ matrix, where all entries of $B$ and $C$ are equal to $\frac{mn}{m+n}$. Let $O$ be a zero matrix of order $m\times m$ and $O^{\prime}$ be a zero matrix of order $n\times n$. Then

[TABLE]

That is,

[TABLE]

Hence

[TABLE]

Therefore

[TABLE]

The proof is complete. ∎

Following corollary is an easy consequence of Theorem 3.2 .

Corollary 3.2.1**.**

$E_{{\rm ISI}}(S_{n})=\frac{2(n-1)^{\frac{3}{2}}}{n}$.

Theorem 3.3**.**

$E_{{\rm ISI}}(K_{n})=(n-1)^{2}$.

Proof.

Since each vertex of $K_{n}$ has degree $n-1$, for $v_{i},v_{j}\in V(K_{n})$ with $i\neq j$, we have

[TABLE]

Hence

[TABLE]

Consequently

[TABLE]

Now

[TABLE]

∎

Remark 3.4**.**

By Theorem 2.3 and Theorem 3.3, it is easily seen that $E_{{\rm ISI}}(\overline{K}_{m,n})=m^{2}+n^{2}-2(m+n-1)$.

A graph whose every vertex has equal degree is called a regular graph. A graph whose every vertex has degree $k$ is called a $k$-regular graph. Zangi et al. [26] prove the following result.

Theorem 3.5** (Zangi et al. [26]).**

Suppose $G$ is an $n$-vertex $k$-regular graph. Then $E_{{\rm ISI}}(G)=\frac{k}{2}~{}E(G)$.

4 Bounds and integral representation for ISI energy

In this section, we give some bounds for the ISI energy of graphs.

Let $B$ is a matrix of order $n\times n$ such that $b_{ij}=0$ if $v_{i}\not\sim v_{j}$ and $b_{ij}=\mathcal{F}(d_{i},d_{j})$ if $v_{i}\sim v_{j}$, where $\mathcal{F}$ is the function with the property $\mathcal{F}(y,z)=\mathcal{F}(z,y)$. Das et al. [8] prove the following theorem for eigenvalues of degree based energies of graphs.

Theorem 4.1** (Das et al. [8]).**

For the eigenvalues $f_{1}\geq f_{2}\geq\dots\geq f_{n}$ of a matrix $M$, the followimg inequalities hold.

[TABLE]

for $k=2,\dots,n-1$.

The following result is obtained by using Theorem 4.1,.

Theorem 4.2**.**

For the eigenvalues $\tau_{1}\geq\tau_{2}\geq\dots\geq\tau_{n}$ of $S(G)$, the following inequalities hold.

[TABLE]

for $k=2,\dots,n$.

Using Theorem 2.2 and Theorem 4.2, we get the following result for an $n$-vertex connected graph $G$.

Theorem 4.3**.**

For $S$-eigenvalues $\tau_{1}\geq\tau_{2}\geq\dots\geq\tau_{n}$ of a connected graph $G$, the following inequalities hold.

[TABLE]

for $k=2,\dots,n-1$.

In next theorem, we find bounds for ISI energy in terms of trace of matrix $S^{2}(G)$ and determinant of $S(G)$.

Theorem 4.4**.**

Let $G$ be an $n$-vertex simple graph, $n\geq 2$. Then

[TABLE]

where $\Theta=\det S(G)$

Proof.

As we know that arithmetic mean is always less than quadratic mean, therefore

[TABLE]

Arithmetic-quadratic mean inequality gives,

[TABLE]

The proof is complete. ∎

Now we have the following theorem. The proof is same as the proof of Theorem $3$ [8] and is thus excluded.

Theorem 4.5**.**

Let $G$ be a simple $n$-vertex graph with $n\geq 2$ vertices. Then

[TABLE]

In Theorem $4.8$, we obtain bounds for ISI energy in terms of number of edges, minimum and maximum degrees of a simple graph.

Theorem 4.6**.**

Suppose $G$ is an $n$-vertex simple graph with $m$ edges, minimum degree $\delta$ and maximum degree $\Delta$. Then

[TABLE]

Proof.

For each vertex $v_{i}$ of $G$, $\delta\leq d_{i}\leq\Delta$, $i=1,2,\dots n$. Using this fact, we get

[TABLE]

[TABLE]

Hence

[TABLE]

[TABLE]

Now using Theorem 4.5, we obtain the desired result. ∎

Coulson [5] prove the following integral representation of energy of graphs.

Theorem 4.7** (Coulson [5]).**

Let $G$ be an $n$-vertex simple graph, then

[TABLE]

where $\Phi^{\prime}(G,\lambda)=\frac{d}{d\lambda}\Phi(G,\lambda)$ and $\dot{\iota}=\sqrt{-1}$.

Next theorem is an analogue of Theorem 4.7.

Theorem 4.8**.**

Let $G$ be a simple graph of order $n$. Then

[TABLE]

where $\Phi^{\prime}_{S}(G,\lambda)=\frac{d}{d\lambda}\Phi_{S}(G,\lambda)$ and $\dot{\iota}=\sqrt{-1}$.

Corollary 4.8.1**.**

If $G$ is an $n$-vertex graph then

[TABLE]

The following result is similar to the graph energy.

Theorem 4.9**.**

Let $G$ be an $n$-vertex graph with $S$-characteristic polynomial $\Phi_{S}(G,\lambda)=\lambda^{n}+\sum_{i=1}^{n}b_{i}\lambda^{n-i}.$ Then

[TABLE]

The following result is based on our numerical testing. The application of Coulson-type integral expression for proving the conjecture was (so far) not successful.

Conjecture 4.10**.**

Among all $n$-vertex tress, the tree with minimal ISI energy is $S_{n}$ and the tree with maximal ISI energy is $P_{n}$, where $n\geq 2$.

5 S-Equienergetic graphs

Two graphs with same $S$-spectrum are said to be $S$-cospectral, otherwise $S$-noncospectral. Two S-equienergetic graphs of same order are the graphs which have same ISI energy. Two isomorphic graphs are always $S$-cospectral and thus are S-equienergetic. We construct few classes of $S$-noncospectral $S$-equienergetic graphs.

The line graph, denoted by $L(G)$, of $G$, is the graph with $V(L(G))=E(G)$ and two vertices of $L(G)$ are connected by an edge if edges incident on it are adjacent in $G$.

Let $G$ be $k$-regular $n$-vertex graph. Let $L_{1}(G)=L(G)$, $L_{i}(G)=L(L_{i-1}(G))$, $i=1,2,\dots$, be the iterated line graphs of $G$. Ramane et al. [22] prove the following energy formula for $L_{2}(G)$.

[TABLE]

Theorem 5.1**.**

Suppose $G_{1}$ and $G_{2}$ are two $k$-regular $n$-vertex $A$-noncospectral graphs. Then $L_{2}(G_{1})$ and $L_{2}(G_{2})$ are $S$-noncospectral $S$-equienergetic graphs.

Proof.

If $G$ is an $n$-vertex $k$-regular then $L_{2}(G)$ is $\frac{1}{2}nk(k-1)$-vertex $(4k-6)$-regular graph. By Theorem 3.5 and (5.5), we get

[TABLE]

Hence $E_{{\rm ISI}}(L_{2}(G_{1}))=E_{{\rm ISI}}(L_{2}(G_{2}))$.

Since $S(L_{2}(G))=(2k-3)A(L_{2}(G))$ and $L_{2}(G_{1})$ and, $L_{2}(G_{2})$ are $A$-noncospectral graphs, therefore $L_{2}(G_{1})$ and $L_{2}(G_{2})$ are also $S$-noncospectral graphs. ∎

Corollary 5.1.1**.**

Suppose $G_{1}$ and $G_{2}$ are two $k$-regular $n$-vertex $A$-noncospectral graphs. Then for any $m\geq 2$, $L_{m}(G_{1})$ and $L_{m}(G_{2})$ are $S$-noncospectral $S$-equienergetic.

Theorem 5.2**.**

Suppose $G_{1}$ and $G_{2}$ are two $n$-vertex $S$-noncospectral $S$-equienergetic graphs. Then $G_{1}\cup\overline{K}_{r}$ and $G_{2}\cup\overline{K}_{r}$ are $S$-noncospectral $S$-equienergetic.

Proof.

By Theorem 2.3, we have

[TABLE]

Since $G_{1}$ and $G_{2}$ are $S$-noncospectral, therefore $G_{1}\cup\overline{K}_{r}$ and $G_{2}\cup\overline{K}_{r}$ are $S$-noncospectral. ∎

Corollary 5.2.1**.**

Suppose $G_{1}$ and $G_{2}$ are two $k$-regular $n$-vertex $A$-noncospectral graphs. Then for any $m\geq 2$, $L_{m}(G_{1})\cup\overline{K}_{r}$ and $L_{m}(G_{1})\cup\overline{K}_{r}$ are $S$-noncospectral $S$-equienergetic.