The Degree Energy of a Graph

Abstract

The incidence of edges on vertices is a cornerstone of graph theory, with profound implications for various graph properties and applications. Understanding degree distributions and their implications is crucial for analyzing and modeling real-world networks. This study investigates the impact of vertex degree distribution on the energy landscape of graphs in network theory. By analyzing how vertex connectivity influences graph energy, the research enhances the understanding of network structure and dynamics. It establishes important properties and sharp bounds related to degree spectra and degree energy. Furthermore, the study determines the degree spectra and degree energy for several key families of graphs, providing valuable insights with potential applications across various fields.

1. Introduction

In recent decades, spectral graph theory has rapidly advanced, leading to the emergence of various new sub-fields within graph theory. The adjacency matrix of a graph allowed for the investigation of the degree of its vertices. This work opened up many new perspectives on the use of matrices in the study of graphs. A number of previously established graph theoretical concepts, including matching and chromatic number [1], were examined in relation to the spectral radius, the second largest eigenvalue, the least eigenvalue, and the absolute sum of these eigenvalues [2]. Because of its practical relevance, the energy and spectral radius also captured the interest of chemists and computer scientists [3]. Numerous graph polynomials have been defined on various graph matrices such as adjacency matrix, Laplacian matrix, degree sum matrix, distance matrix, skew-adjacency matrix, randic matrix, minimum covering matrix, and labelled matrix [4,5,6,7,8,9,10] in the literature.

The energy of the adjacency matrix of a graph has since received a lot of attention [11,12]. It is well-established that the $\pi$-electron energy of a molecule can be approximately estimated using graph energy, linking theoretical chemistry closely with graph energy concepts [13,14,15]. In 1971, McClelland [16] provided both lower and upper bounds for $\pi$-electron energy. Hosamani et al. [17] described the degree sum energy of graphs and established some lower bounds for this energy. Graph energy and its applications were covered by Gutman et al. [18], who included information on approximately 100 different types of graph energies and their uses in various fields. Further, Pirzada et al. [19] addressed the specific property that the energy of a graph cannot be the square root of an odd integer. Wilson et al. [20] offered a foundation for the fundamental concepts and principles of the field by discussing on various graph properties and their applications. Moreover, Zhang et al. [21] explored the spectral radii associated with the maximum and minimum degrees in operations by providing a detailed analysis and results related to the spectral properties of graphs. Xiaolong Shi et al. [22] expanded the energy concept on the picture fuzzy graph and sought to use the concept in solving problems on the neutrality state. For additional information and a general understanding of graph spectra and graph energy, one can refer to [23,24,25,26,27].

The objective of this study is to introduce and analyze the degree matrix of a graph, emphasizing the relationship between vertex degrees and adjacency. The study aims to uncover key properties of degree spectra and degree energy, and to derive analytical expressions for the degree energy of several significant families of graphs. This approach builds on previous research by expanding the understanding of how degree-related metrics influence graph energy and spectral properties.

For notational convenience and better understanding, we use the following notations in our study.

 Notations:

 

$z_s$,

is the total number of neighbouring vertex pairs with the same degree.

$z_d$,

is the total number of neighbouring vertex pairs with different degrees.

$z_n$,

is the total number of non-neighboring vertex pairs with identical degrees.

Let $G=(V,E)$ be a finite, undirected, simple graph with order k and size l. The number of edges incident to a vertex $v\in V\left(G\right)$ is its degree, $d\left(v\right)$. The $k\times k$ matrix that represents the degree matrix of a graph G is defined by $M_D\left(G\right)=\left(m_{rs}\right)$, where,

$$m_{rs}=\left\{\begin{array}{cc}2\hfill & \mathrm{if}{v}_r\mathrm{and}{v}_s\mathrm{are}\mathrm{adjacent}\mathrm{with}d\left(v_r\right)=d\left(v_s\right),\hfill \\ 1\hfill & \mathrm{if}{v}_r\mathrm{and}{v}_s\mathrm{are}\mathrm{adjacent}\mathrm{with}d\left(v_r\right)\ne d\left(v_s\right),\hfill \\ -1\hfill & \mathrm{if}{v}_r\mathrm{and}{v}_s\mathrm{are}\text{non-adjacent}\mathrm{with}d\left(v_r\right)=d\left(v_s\right),\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

The set of eigenvalues ${\lambda }_1\ge {\lambda }_2\ge \dots \ge {\lambda }_r$ with their algebraic multiplicities $w_1,w_2,\cdots ,w_r$ of $M_D\left(G\right)$ is called a degree spectra of G, denoted by $Spe{c}_D\left(G\right)$ and represented as follows,

$$Spe{c}_D\left(G\right)=\left(\begin{array}{cccc}{\lambda }_1& {\lambda }_2& \cdots & {\lambda }_r\\ w_1& w_2& \cdots & w_r\end{array}\right)$$

We consider the degree energy as the sum of absolute degree eigenvalues, i.e.,

$$E_D\left(G\right)=\sum _{i=1}^r{w}_i|{\lambda }_i|$$

Example 1.

In Figure 1, vertices $v_1,v_2,v_3,v_5,$ and $v_6$ have degree 4, while $v_4$ and $v_7$ have degree 2.

We obtain the degree matrix as shown below,

$$M=\begin{array}{cc} & \begin{array}{ccccccc}{v}_1& v_2& v_3& v_4& v_5& v_6& v_7\end{array}\\ \begin{array}{c}\hfill v_1\\ \hfill v_2\\ \hfill v_3\\ \hfill v_4\\ \hfill v_5\\ \hfill v_6\\ \hfill v_7\end{array}& \left[\begin{array}{ccccccc}0& 2& 2& 0& 2& 2& 0\\ 2& 0& 2& 1& -1& 2& 0\\ 2& 2& 0& 0& 2& -1& 1\\ 0& 1& 0& 0& 0& 1& -1\\ 2& -1& 2& 0& 0& 2& 1\\ 2& 2& -1& 1& 2& 0& 0\\ 0& 0& 1& -1& 1& 0& 0\end{array}\right]\end{array}$$

The characteristic equation for the above obtained matrix is ${\lambda }^7-39{\lambda }^5-24{\lambda }^4+255{\lambda }^3+88{\lambda }^2-201\lambda -80=0$. The vertex degree eigenvalues are ${\lambda }_1=1$, ${\lambda }_2=2.4142$, ${\lambda }_3=5.9680$, ${\lambda }_4=-0.4142$, ${\lambda }_5=-0.8636$, ${\lambda }_6=-3.1044$, ${\lambda }_7=-5$. Therefore vertex degree energy is $E_D=18.7644$.

2. Main Results

2.1. Basic Properties on Degree Spectra and Degree Energy of a Graph

Let G be a degree graph with k vertices and l edges, represented by its degree matrix $M_D\left(G\right)$. The subsequent theorems establish the properties of the degree energy $E_D\left(G\right)$, including the sum of the squared eigenvalues, using the minimum eigenvalue, upper and lower bounds for $E_D\left(G\right)$, bounds for the maximum absolute degree eigenvalue, and the sum of the largest and smallest eigenvalues.

Theorem 1.

Let ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_k$ be the vertex degree eigenvalues of $M_D\left(G\right)$ then,

$$\sum _{i=1}^k{\lambda }_i^2=2[4z_s+z_d+z_n]$$

Proof. 

The trace of $M_D^2\left(G\right)$ is equal to the sum of the squares of the eigenvalues of $M_D\left(G\right)$,

$$\begin{array}{cc}\hfill \sum _{i=1}^k{\lambda }_i^2& =\sum _{i=1}^k\sum _{j=1}^k{a}_{ij}{a}_{ji}\hfill \\ & =2\sum _{i<j}^k{\left(a_{ij}\right)}^2\hfill \\ \hfill & =2[4z_s+z_d+z_n]\hfill \end{array}$$

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Theorem 2.

Given a graph G, if $\mu =min\{|{\lambda }_1|,|{\lambda }_2|,\dots ,|{\lambda }_k|\}$then , $\mu \le |det\left(M_D\left(G\right)\right){|}^{{\displaystyle \frac{1}{k}}}$

Proof. 

$$\begin{array}{cc}\hfill |{\lambda }_1||{\lambda }_2|\dots |{\lambda }_k|& =|det(M_D\left(G\right))|.\hfill \\ \hfill \mu \mu \mu \dots \mu & \le |det(M_D\left(G\right))|\hfill \\ \hfill \Rightarrow \mu & \le |det\left(M_D\left(G\right)\right){|}^{{\displaystyle \frac{1}{k}}}\hfill \end{array}$$

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Theorem 3.

Let G represent a graph. Then,

$$\sqrt{2[4z_s+z_d+z_n]+k(k-1)D^{{\displaystyle \frac{2}{k}}}}\le E_D\left(G\right)\le \sqrt{2k(4z_s+z_d+z_n)}$$

Proof. 

Through the Cauchy–Schwarz inequality,

$$\begin{array}{cc}\hfill {\left[E_D\left(G\right)\right]}^2& ={\left(\sum _{j=1}^k|{\lambda }_j|\right)}^2\hfill \\ \hfill & \le k\sum _{j=1}^k{|{\lambda }_j|}^2\hfill \\ \hfill & =k\sum _{j=1}^k{\lambda }_j^2\hfill \\ \hfill & =2k(4z_s+z_d+z_n)\hfill \\ \hfill E_D\left(G\right)& \le \sqrt{2k(4z_s+z_d+z_n)}\hfill \end{array}$$

Now, let us consider,

$$\begin{array}{cc}\hfill \left[E_{D}G\right){]}^2& ={\left(\sum _{i=1}^k|{\lambda }_i|\right)}^2\hfill \\ & =\left(\sum _{i=1}^k|{\lambda }_i|\right)\left(\sum _{j=1}^k|{\lambda }_j|\right)\hfill \\ & =\sum _{i=1}^k|{\lambda }_i{|}^2+\sum _{i\ne j}^k|{\lambda }_i||{\lambda }_j|\hfill \end{array}$$

The geometric mean inequality and the arithmetic mean allow us to

$$\begin{array}{cc}\hfill {\left[E_D\left(G\right)\right]}^2\ge & \sum _{i=1}^k{|{\lambda }_i|}^2+k(k-1){\left(\prod _{i\ne j}|{\lambda }_i||{\lambda }_j|\right)}^{{\displaystyle \frac{1}{k(k-1)}}}\hfill \\ \hfill \ge & \sum _{i=1}^k{|{\lambda }_i|}^2+k(k-1){\left(\prod _{i=1}^k{|{\lambda }_i|}^{2(r-1)}\right)}^{{\displaystyle \frac{1}{k(k-1)}}}\hfill \\ \hfill =& \sum _{i=1}^r{|{\lambda }_i|}^2+k(k-1)|\prod _{i=1}^k{\lambda }_i{|}^{{\displaystyle \frac{2}{k}}}\hfill \\ \hfill =& 2[4z_s+z_d+z_n]+k(k-1)D^{{\displaystyle \frac{2}{k}}}\hfill \\ \hfill [E_D\left(G\right)]\ge & \sqrt{2[4z_s+z_d+z_n]+k(k-1)D^{{\displaystyle \frac{2}{k}}}}\hfill \end{array}$$

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Theorem 4.

(Bounds for maximum absolute degree eigenvalue) If the vertex degree eigenvalue of G is ${\mu }_1\left(G\right)={max}_{i\le i\le r}\{|{\lambda }_i|\}$, then

$$\sqrt{{\displaystyle \frac{2(4z_s+z_d+z_n)}{k}}}\le {\mu }_1\left(G\right)\le \sqrt{2(4z_s+z_d+z_n)}$$

Proof. 

Consider that,

$$\begin{array}{cc}\hfill {\mu }_1^2\left(G\right)& =\underset{i\le i\le r}{max}\{|{\lambda }_i{|}^2\}\hfill \\ \hfill & \le \sum _{i=1}^k{|{\lambda }_i|}^2\hfill \\ \hfill & =2[4z_s+z_d+z_n]\hfill \\ \hfill {\mu }_1^2\left(G\right)& \le \sqrt{2[4z_s+z_d+z_n]}\hfill \end{array}$$

Now, let us consider,

$$\begin{array}{cc}\hfill k{\mu }_1^2\left(G\right)& \ge \sum _{i=1}^k{|{\lambda }_i|}^2\hfill \\ \hfill & \ge 2[4z_s+z_d+z_n]\hfill \\ \hfill {\mu }_1^2\left(G\right)& \ge \sqrt{{\displaystyle \frac{2[4z_s+z_d+z_n]}{k}}}\hfill \end{array}$$

$$\therefore \sqrt{{\displaystyle \frac{2(4z_s+z_d+z_n)}{k}}}\le {\mu }_1\left(G\right)\le \sqrt{2(4z_s+z_d+z_n)}$$

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Theorem 5.

Consider a graph G with $k\ge 3$ vertices. Assessing the greatest and smallest eigenvalues of a graph is ${\lambda }_1$ and ${\lambda }_k$, respectively, then,

$$({\lambda }_1+{\lambda }_k)\le 2\sqrt{{\displaystyle \frac{(k-2)(4z_s+z_d+z_n)}{k}}}$$

Proof. 

For ${\lambda }_1\ge {\lambda }_2\ge \dots \ge {\lambda }_k$ of G, from the results of Theorem 1. Applying the Schwarz–Cauchy inequality,

$$\begin{array}{cc}\hfill {\left(\sum _{i=2}^{k-1}{\lambda }_i\right)}^2& \le \left(\sum _{i=2}^{k-1}1\right)\left(\sum _{i=2}^{k-1}{\lambda }_i^2\right)\hfill \\ \hfill {(-{\lambda }_1-{\lambda }_k)}^2& \le (k-2)[2(4z_s+z_d+z_n)-{\lambda }_1^2-{\lambda }_k^2]\hfill \\ \hfill & =2(k-2)(4z_s+z_d+z_n)-(k-2)({\lambda }_1^2+{\lambda }_k^2)\hfill \\ \hfill \therefore 2(k-2)(4z_s+z_d+z_n)& \ge {({\lambda }_1+{\lambda }_k)}^2+(k-2)({\lambda }_1^2+{\lambda }_k^2)\hfill \\ \hfill & ={({\lambda }_1+{\lambda }_k)}^2+(k-2)({({\lambda }_1+{\lambda }_k)}^2-2{\lambda }_1{\lambda }_k)\hfill \\ \hfill & ={({\lambda }_1+{\lambda }_k)}^2(k-1)-2(k-2){\lambda }_1{\lambda }_k\hfill \\ \hfill Anyway,{\left({\displaystyle \frac{{\lambda }_1+{\lambda }_k}{2}}\right)}^2& \ge {\lambda }_1{\lambda }_k\hfill \\ \hfill \Rightarrow -\left({\lambda }_1{\lambda }_k\right)& \ge -{\left({\displaystyle \frac{{\lambda }_1+{\lambda }_k}{2}}\right)}^2\hfill \end{array}$$

$$\begin{array}{cc}\hfill Thus,2(k-2)(4z_s+z_d+z_n)& \ge {({\lambda }_1+{\lambda }_k)}^2(k-1)-2(k-2){\left({\displaystyle \frac{{\lambda }_1+{\lambda }_k}{2}}\right)}^2\hfill \\ \hfill & ={({\lambda }_1+{\lambda }_k)}^2{\displaystyle \frac{k}{2}}\hfill \\ \hfill Hence,{\displaystyle \frac{4(k-2)(4z_s+z_d+z_n)}{k}}& \ge {({\lambda }_1+{\lambda }_k)}^2\hfill \\ \hfill ({\lambda }_1+{\lambda }_k)& \le 2\sqrt{{\displaystyle \frac{(k-2)(4z_s+z_d+z_n)}{k}}}\hfill \end{array}$$

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2.2. Degree Spectra and Degree Energy of a Graph

This section and the following one focus on the energy associated with vertex degrees in graphs. We introduce the concept of the complement of a degree graph and subsequently calculate the degree energy for specific families of graphs and their corresponding complements.

Theorem 6.

The degree energy of a complete graph $K_r$ is $4(r-1)$.

Proof. 

The degree matrix for a complete graph $K_r$ is

$$M_D\left(K_r\right)={\left[\begin{array}{ccccc}0& 2& 2& \dots & 2\\ 2& 0& 2& \dots & 2\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 2& 2& 2& \dots & 0\end{array}\right]}_{r\times r}$$

Then, the characteristic polynomial of degree matrix is as follows

$${(\lambda +2)}^{r-1}(\lambda -2(r-1))=0$$

The degree spectra of a complete graph $k_r$ is

$$Spe{c}_D\left(K_r\right)=\left(\begin{array}{cc}2(r-1)& -2\\ 1& r-1\end{array}\right)$$

Therefore, the degree energy of a complete graph $k_r$ is,

$$E_D\left(K_r\right)=4(r-1).$$

□

Theorem 7.

The degree energy of a star graph $K_{1,r}$ is $2r.$

Proof. 

The degree matrix of a star graph $K_{1,r}$ is

$$M_D\left(K_{1,r}\right)={\left[\begin{array}{ccccc}0& 1& 1& \dots & 1\\ 1& 0& -1& \dots & -1\\ 1& -1& 0& \dots & -1\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& -1& -1& \dots & 0\end{array}\right]}_{(r+1)\times (r+1)}$$

Then, the characteristic polynomial of the degree matrix is

$$(\lambda +r){(\lambda -1)}^r=0$$

And the degree spectra are

$$Spe{c}_D\left(K_{1,r}\right)=\left(\begin{array}{cc}-r& 1\\ 1& r\end{array}\right)$$

Hence, the degree energy of a star graph is,

$$E_D\left(K_{1,r}\right)=2r.$$

□

Theorem 8.

The degree energy of a double star graph $S_{r,r}$ is

$$\sqrt{4r^2+8r+1}+\sqrt{4r+9}+2r-2.$$

Proof. 

Let $S_{r,r}$ be a double star graph and its degree matrix is

$$M_D\left(S_{r,r}\right)={\left[\begin{array}{cccccccccc}0& 1& 1& \dots & 1& 2& 0& 0& \dots & 0\\ 1& 0& -1& \cdots & -1& 0& -1& -1& \cdots & -1\\ 1& -1& 0& \cdots & -1& 0& -1& -1& \cdots & -1\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& -1& -1& \cdots & 0& 0& -1& -1& \cdots & -1\\ 2& 0& 0& \cdots & 0& 0& 1& 1& \cdots & 1\\ 0& -1& -1& \cdots & -1& 1& 0& -1& \cdots & -1\\ 0& -1& -1& \cdots & -1& 1& -1& 0& \cdots & -1\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& -1& -1& \cdots & -1& 1& -1& -1& \cdots & 0\end{array}\right]}_{2r\times 2r}$$

Then, the characteristic polynomial of the degree matrix of $S_{r,r}$ is,

$$({\lambda }^2+\lambda -(r+2))({\lambda }^2+(2r-3)\lambda -(5r-2)){(\lambda -1)}^{2(r-1)}=0$$

And the degree spectra of $S_{r,r}$ is as follows

$$Spe{c}_D\left(S_{r,r}\right)=\left(\begin{array}{ccc}{\displaystyle \frac{-1}{2}}\pm {\displaystyle \frac{\sqrt{4r+9}}{2}}& {\displaystyle \frac{-(2r-3)}{2}}\pm {\displaystyle \frac{\sqrt{4r^2+8r+1}}{2}}& 1\\ 1& 1& 2(r-1)\end{array}\right)$$

Thus, the degree energy of the double star graph $S_{r,r}$ can be expressed in the form,

$$E_D\left(S_{r,r}\right)=\sqrt{4r+9}+\sqrt{4r^2+8r+1}+2r-2.$$

□

Theorem 9.

The degree energy of a crown graph $S_r^0$ is $10r-12$.

Proof. 

Let $S_r^0$ be a crown graph and its degree matrix is

$$M_D\left(S_r^0\right)={\left[\begin{array}{cccccccccc}0& -1& -1& \dots & -1& -1& 2& 2& \dots & 2\\ -1& 0& -1& \cdots & -1& 2& -1& 2& \cdots & 2\\ -1& -1& 0& \cdots & -1& 2& 2& -1& \cdots & 2\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ -1& -1& -1& \cdots & 0& 2& 2& 2& \cdots & -1\\ -1& 2& 2& \cdots & 2& 0& -1& -1& \cdots & -1\\ 2& -1& 2& \cdots & 2& -1& 0& -1& \cdots & -1\\ 2& 2& -1& \cdots & 2& -1& -1& 0& \cdots & -1\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 2& 2& 2& \cdots & -1& -1& -1& -1& \cdots & 0\end{array}\right]}_{2r\times 2r}$$

Then, the characteristic equation of the degree matrix of $S_r^0$ can be obtained in the form,

$$(\lambda +(3r-4))(\lambda -(r-2)){(\lambda +2)}^{r-1}{(\lambda -4)}^{r-1}=0$$

Hence, the degree spectra of $S_r^0$ can be expressed as

$$Spe{c}_D\left(S_r^0\right)=\left(\begin{array}{cccc}-(3r-4)& r-2& -2& 4\\ 1& 1& r-1& r-1\end{array}\right)$$

Therefore, the degree energy of the crown graph can be computed as follows,

$$E_D\left(S_r^0\right)=10r-12.$$

□

Theorem 10.

The degree energy of the Moore graph $K_{r,r}$ is $6r-2$.

Proof. 

Let $K_{r,r}$ represent the Moore graph and its degree matrix can be expressed as follows

$$M_D\left(\overline{K_{r,r}}\right)={\left[\begin{array}{cccccccccc}0& -1& -1& \dots & -1& 2& 2& 2& \dots & 2\\ -1& 0& -1& \cdots & -1& 2& 2& 2& \cdots & 2\\ -1& -1& 0& \cdots & -1& 2& 2& 2& \cdots & 2\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ -1& -1& -1& \cdots & 0& 2& 2& 2& \cdots & 2\\ 2& 2& 2& \cdots & 2& 0& -1& -1& \cdots & -1\\ 2& 2& 2& \cdots & 2& -1& 0& -1& \cdots & -1\\ 2& 2& 2& \cdots & 2& -1& -1& 0& \cdots & -1\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 2& 2& 2& \cdots & 2& -1& -1& -1& \cdots & 0\end{array}\right]}_{2r\times 2r}$$

Then, the characteristic equation of $K_{r,r}$ can be considered as,

$$(\lambda +(3r-1))(\lambda -(r+1)){(\lambda -1)}^{2r-2}=0$$

Therefore, the degree spectra of $K_{r,r}$ is

$$Spe{c}_D\left(K_{r,r}\right)=\left(\begin{array}{ccc}-(3r-1)& r+1& 1\\ 1& 1& 2r-2\end{array}\right)$$

Hence, the degree energy of the Moore graph is

$$E_D\left(K_{r,r}\right)=6r-2.$$

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Theorem 11.

For any integer $r\ge 2$, the degree energy of a cocktail party graph $K_{r\times 2}$ is $10(r-1)$.

Proof. 

Let $K_{r\times 2}$ be the cocktail party graph; then, its degree matrix can be expressed in the form

$$M_D\left(K_{r\times 2}\right)={\left[\begin{array}{cccccccccc}0& 2& 2& \dots & 2& -1& 2& 2& \dots & 2\\ 2& 0& 2& \cdots & 2& 2& -1& 2& \cdots & 2\\ 2& 2& 0& \cdots & 2& 2& 2& -1& \cdots & 2\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 2& 2& 2& \cdots & 0& 2& 2& 2& \cdots & -1\\ -1& 2& 2& \cdots & 2& 0& 2& 2& \cdots & 2\\ 2& -1& 2& \cdots & 2& 2& 0& 2& \cdots & 2\\ 2& 2& -1& \cdots & 2& 2& 2& 0& \cdots & 2\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 2& 2& 2& \cdots & -1& 2& 2& 2& \cdots & 0\end{array}\right]}_{2r\times 2r}$$

Then, the characteristic polynomial of $K_{r\times 2}$ is,

$$(\lambda -(4r-5)){(\lambda +5)}^{r-1}{(\lambda -1)}^r=0$$

Thus, the degree spectra of $K_{r\times 2}$ can be represented as follows,

$$Spe{c}_D\left(K_{r\times 2}\right)=\left(\begin{array}{ccc}4r-5& -5& 1\\ 1& r-1& r\end{array}\right)$$

Therefore, the degree energy of the cocktail party graph can be computed as follows,

$$E_D\left(K_{r\times 2}\right)=10(r-1).$$

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Theorem 12.

For any integer $r\ge 2$, the degree energy of $K_{2,r}$ is $2(r+1)$.

Proof. 

Let $K_{2,r}$ be a complete bipartite graph and its degree matrix is as follows

$$M_D\left(K_{2,r}\right)={\left[\begin{array}{cccccccccc}0& 1& 1& \dots & 1& 1& 1& 1& \dots & -1\\ 1& 0& -1& \cdots & -1& -1& -1& -1& \cdots & 1\\ 1& -1& 0& \cdots & -1& -1& -1& -1& \cdots & 1\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& -1& -1& \cdots & 0& -1& -1& -1& \cdots & 1\\ 1& -1& -1& \cdots & -1& 0& -1& -1& \cdots & 1\\ 1& -1& -1& \cdots & -1& -1& 0& -1& \cdots & 1\\ 1& -1& -1& \cdots & -1& -1& -1& 0& \cdots & 1\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ -1& 1& 1& \cdots & 1& 1& 1& 1& \cdots & 0\end{array}\right]}_{(r+2)\times (r+2)}$$

Then, the characteristic equation of $K_{2,r}$ is,

$$(\lambda +(r+1)){(\lambda -1)}^{r+1}=0$$

Hence, the degree spectra of $K_{2,r}$ can be considered as,

$$Spe{c}_D\left(K_{2,r}\right)=\left(\begin{array}{cc}-(r+1)& 1\\ 1& r+1\end{array}\right)$$

Therefore, the degree energy of $K_{2,r}$ is,

$$E_D\left(K_{2,r}\right)=2(r+1).$$

□

Theorem 13.

For any integer $r\ge 2$, the degree energy of a friendship graph $F_r$ is $2\sqrt{r^2-2r+4}+6r-4$.

Proof. 

Let $F_r$ represent a friendship graph and its degree matrix is as follows

$$M_D\left(F_r\right)={\left[\begin{array}{ccccccccccc}0& 1& 1& 1& \dots & 1& 1& 1& 1& \dots & 1\\ 1& 0& -1& -1& \cdots & -1& 2& -1& -1& \cdots & -1\\ 1& -1& 0& -1& \cdots & -1& -1& 2& -1& \cdots & -1\\ 1& -1& -1& 0& \cdots & -1& -1& -1& 2& \cdots & -1\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& -1& -1& -1& \cdots & 0& -1& -1& -1& \cdots & 2\\ 1& 2& -1& -1& \cdots & -1& 0& -1& -1& \cdots & -1\\ 1& -1& 2& -1& \cdots & -1& -1& 0& -1& \cdots & -1\\ 1& -1& -1& 2& \cdots & -1& -1& -1& 0& \cdots & -1\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& & ⋮\ddots & ⋮\\ 1& -1& -1& -1& \cdots & 2& -1& -1& -1& \cdots & 0\end{array}\right]}_{2r\times 2r}$$

Then, the characteristic equation of $F_r$ is

$$({\lambda }^2+2(r-2)\lambda -2r){(\lambda -4)}^{r-1}{(\lambda +2)}^r=0$$

Thus, the degree spectra of $F_r$ can be expressed in the form

$$Spe{c}_D\left(F_r\right)=\left(\begin{array}{ccc}-(r-2)\pm \sqrt{r^2-2r+4}& 4& -2\\ 1& r-1& r\end{array}\right)$$

Therefore, the degree energy of friendship graph is

$$E_D\left(F_r\right)=2\sqrt{r^2-2r+4}+6r-4.$$

□

2.3. Degree Spectra and Degree Energy of Complement of a Graph

Theorem 14.

The degree energy of the complement of a star graph $\overline{K_{1,r}}$ is $4(r-1)$.

Proof. 

Let $\overline{K_{1,r}}$ be a complement of a star graph and its degree matrix is

$$M_D\left(\overline{K_{1,r}}\right)={\left[\begin{array}{ccccc}0& 0& 0& \dots & 0\\ 0& 0& 2& \dots & 2\\ 0& 2& 0& \dots & 2\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 2& 2& \dots & 0\end{array}\right]}_{r+1\times r+1}$$

Then, the characteristic equation of $\overline{K_{1,r}}$ is

$$\lambda (\lambda -(2r-2)){(\lambda +2)}^{r-1}=0$$

Thus, the degree spectra of $\overline{K_{1,r}}$ is

$$Spe{c}_D\left(\overline{K_{1,r}}\right)=\left(\begin{array}{ccc}0& 2r-2& -2\\ 1& 1& r-1\end{array}\right)$$

Therefore, the degree energy of a complement of a star graph is,

$$E_D\left(\overline{K_{1,r}}\right)=4(r-1).$$

□

Theorem 15.

The degree energy of complement of a double star graph $\overline{S_{r,r}}$ is

$$\sqrt{16r^2-4r+1}+\sqrt{4r+9}+4(r-1).$$

Proof. 

Let $\overline{S_{r,r}}$ be a complement of a double star graph and its matrix is

$$M_D\left(\overline{S_{r,r}}\right)={\left[\begin{array}{cccccccccc}0& 0& 0& \dots & 0& -1& 1& 1& \dots & 1\\ 0& 0& 2& \cdots & 2& 1& 2& 2& \cdots & 2\\ 0& 2& 0& \cdots & 2& 1& 2& 2& \cdots & 2\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 2& 2& \cdots & 0& 1& 2& 2& \cdots & 2\\ -1& 1& 1& \cdots & 1& 0& 0& 0& \cdots & 0\\ 1& 2& 2& \cdots & 2& 0& 0& 2& \cdots & 2\\ 1& 2& 2& \cdots & 2& 0& 2& 0& \cdots & 2\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& 2& 2& \cdots & 2& 0& 2& 2& \cdots & 0\end{array}\right]}_{2r\times 2r}$$

Then, the characteristic equation of $\overline{S_{r,r}}$ is as follows,

$$({\lambda }^2+\lambda -(r+2))({\lambda }^2-(4r-3)\lambda -(5r-2)){(\lambda +2)}^{2(r-1)}=0$$

Hence, the degree spectra can be expressed as

$$Spe{c}_D\left(\overline{S_{r,r}}\right)=\left(\begin{array}{ccc}{\displaystyle \frac{-1}{2}}\pm {\displaystyle \frac{\sqrt{4r+9}}{2}}& {\displaystyle \frac{(4r-3)}{2}}\pm {\displaystyle \frac{\sqrt{16r^2-4r+1}}{2}}& -2\\ 1& 1& 2(r-1)\end{array}\right)$$

Therefore, the degree energy of a complement of a double star graph is

$$E_D\left(\overline{S_{r,r}}\right)=\sqrt{16r^2-4r+1}+\sqrt{4r+9}+4(r-1).$$

□

Theorem 16.

The degree energy of a complement of a crown graph is $\overline{S_r^0}$ is $10r-10$.

Proof. 

Let $\overline{S_r^0}$ be a complement of a crown graph and its degree matrix is

$$M_D\left(\overline{S_r^0}\right)={\left[\begin{array}{cccccccccc}0& 2& 2& \dots & 2& 2& -1& -1& \dots & -1\\ 2& 0& 2& \cdots & 2& -1& 2& -1& \cdots & -1\\ 2& 2& 0& \cdots & 2& -1& -1& 2& \cdots & -1\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 2& 2& 2& \cdots & 0& -1& -1& -1& \cdots & 2\\ 2& -1& -1& \cdots & -1& 0& 2& 2& \cdots & 2\\ -1& 2& -1& \cdots & -1& 2& 0& 2& \cdots & 2\\ -1& -1& 2& \cdots & -1& 2& 2& 0& \cdots & 2\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ -1& -1& -1& \cdots & 2& 2& 2& 2& \cdots & 0\end{array}\right]}_{2r\times 2r}$$

Then, the characteristic equation of the degree matrix is

$$(\lambda -(3r-5))(\lambda -(r+1)){(\lambda +5)}^{r-1}{(\lambda -1)}^{r-1}=0$$

Hence, the degree spectra of $\overline{S_r^0}$ can be expressed as

$$Spe{c}_D\left(\overline{S_r^0}\right)=\left(\begin{array}{cccc}3r-5& r+1& 1& -5\\ 1& 1& r-1& r-1\end{array}\right)$$

Therefore, the degree energy of a complement of a crown graph is

$$E_D\left(\overline{S_r^0}\right)=10r-10.$$

□

Theorem 17.

The degree energy of a complete bipartite graph $\overline{K_{r,r}}$ is $8(r-1)$.

Proof. 

$\overline{K_{r,r}}$ is a complement of a complete bipartite graph and its matrix is

$$M_D\left(\overline{K_{r,r}}\right)={\left[\begin{array}{cccccccccc}0& 2& 2& \dots & 2& -1& -1& -1& \dots & -1\\ 2& 0& 2& \cdots & 2& -1& -1& -1& \cdots & -1\\ 2& 2& 0& \cdots & -1& -1& -1& -1& \cdots & -1\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 2& 2& 2& \cdots & 0& -1& -1& -1& \cdots & -1\\ -1& -1& -1& \cdots & -1& 0& 2& 2& \cdots & 2\\ -1& -1& -1& \cdots & -1& 2& 0& 2& \cdots & 2\\ -1& -1& -1& \cdots & -1& 2& 2& 0& \cdots & 2\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ -1& -1& -1& \cdots & -1& 2& 2& 2& \cdots & 0\end{array}\right]}_{2r\times 2r}$$

The characteristic equation of $\overline{K_{r,r}}$ is,

$$(\lambda -(3r-2))(\lambda -(r-2)){(\lambda +2)}^{2(r-1)}=0$$

Hence, the spectra of $\overline{K_{r,r}}$ is

$$Spe{c}_D\left(\overline{K_{r,r}}\right)=\left(\begin{array}{ccc}3r-2& r-2& -2\\ 1& 1& 2(r-1)\end{array}\right)$$

Therefore, the energy of a complement of a complete bipartite graph is

$$E_D\left(\overline{K_{r,r}}\right)=8(r-1).$$

□

Theorem 18.

For any integer $r\ge 2$, the degree energy of a complement of a cocktail party graph $\overline{K_{r\times 2}}$ is $8(r-1)$.

Proof. 

Let $\overline{K_{r\times 2}}$ be a complement of a cocktail party graph and its degree matrix be

$$M_D\left(K_{r\times 2}\right)={\left[\begin{array}{cccccccccc}0& -1& -1& \dots & -1& 2& -1& -1& \dots & -1\\ -1& 0& -1& \cdots & -1& -1& 2& -1& \cdots & -1\\ -1& -1& 0& \cdots & -1& -1& -1& 2& \cdots & -1\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ -1& -1& -1& \cdots & 0& -1& -1& -1& \cdots & 2\\ 2& -1& -1& \cdots & -1& 0& -1& -1& \cdots & -1\\ -1& 2& -1& \cdots & -1& -1& 0& -1& \cdots & -1\\ -1& -1& 2& \cdots & -1& -1& -1& 0& \cdots & -1\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ -1& -1& -1& \cdots & 2& -1& -1& -1& \cdots & 0\end{array}\right]}_{2r\times 2r}$$

Now, the characteristic equation is as follows

$$(\lambda +(2r-4)){(\lambda -4)}^{r-1}{(\lambda +2)}^r=0$$

Hence, the degree spectra of $\overline{K_{r\times 2}}$ can be expressed as

$$Spe{c}_D\left(\overline{K_{r\times 2}}\right)=\left(\begin{array}{ccc}-(2r-4)& 4& -2\\ 1& r-1& r\end{array}\right)$$

Therefore, the degree energy of a complement of a cocktail party graph is

$$E_D\left(\overline{K_{r\times 2}}\right)=8(r-1).$$

□

Theorem 19.

For any integer $r\ge 2$, the degree energy of a complement of a friendship graph $\overline{F_r}$ is $10(n-1)$.

Proof. 

Let $\overline{F_r}$ be a complement of a friendship graph and its degree matrix be

$$M_D\left(\overline{F_r}\right)={\left[\begin{array}{ccccccccccc}0& 0& 0& 0& \dots & 0& 0& 0& 0& \dots & 0\\ 0& 0& 2& 2& \cdots & 2& -1& 2& 2& \cdots & 2\\ 0& 2& 0& 2& \cdots & 2& 2& -1& 2& \cdots & 2\\ 0& 2& 2& 0& \cdots & 2& 2& 2& -1& \cdots & 2\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 2& 2& 2& \cdots & 0& 2& 2& 2& \cdots & -1\\ 0& -1& 2& 2& \cdots & 2& 0& 2& 2& \cdots & 2\\ 0& 2& -1& 2& \cdots & 2& 2& 0& 2& \cdots & 2\\ 0& 2& 2& -1& \cdots & 2& 2& 2& 0& \cdots & 2\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& & ⋮\ddots & ⋮\\ 0& 2& 2& 2& \cdots & -1& 2& 2& 2& \cdots & 0\end{array}\right]}_{2r\times 2r}$$

Now, the characteristic equation is as follows

$$\lambda (\lambda -(4r-5)){(\lambda +5)}^{r-1}{(\lambda -1)}^r=0$$

Thus, the degree spectra of $\overline{F_r}$ can be expressed as

$$Spe{c}_D\left(\overline{F_r}\right)=\left(\begin{array}{cccc}0& 4r-5& -5& 1\\ 1& 1& r-1& r\end{array}\right)$$

Therefore, the energy of the complement of Friendship graph is,

$$E_D\left(\overline{F_r}\right)=10(r-1).$$

□

3. Conclusions

The degree distribution in a graph plays an important role in the analysis of the network. In this context, we tried to reveal some important properties that are purely related to the degree distribution using linear algebra. By utilizing the degree matrix, a degree-based representation of a simple graph G, we derived novel spectral results. This approach reduced the dimensionality of the matrices involved in eigenvalue calculations and, in many cases, enabled explicit eigenvalue determination. We introduced a new concept akin to graph energy, termed "degree energy", for which we established bounds and computed the exact energy for specific graph families.