On Maximum Degree and Maximum Reverse Degree Energies of Splitting and Shadow graph of Complete graph

Ibrahim, Arooj; Nazeer, Saima

doi:10.61091/um119-08

Abstract

1. Introduction
2. Preliminaries
3. Maximum Degree Energy of Complete Graph
4. Maximum Degree Energy of Splitting Graph
5. Maximum Degree Energy of Shadow Graph
6. Maximum Reverse Degree Energy of Complete Graph
7. Maximum Reverse Degree Energy of Complete Graph
8. Maximum Reverse Degree Energy of Splitting Graph
9. Maximum Reverse Degree Energy of Shadow Graph
10. Conclusion

References

Utilitas Mathematica

Volume 119
Pages: 73-82

Research article

On Maximum Degree and Maximum Reverse Degree Energies of Splitting and Shadow graph of Complete graph

^¹, ^¹

¹Department of Mathematics, Lahore College for Women University, Lahore-Pakistan

Received: 12/08/2023
Accepted: 07/12/2023
Published: 31/03/2024

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Abstract

In this paper, the relations of maximum degree energy and maximum reserve degree energy of a complete graph after removing a vertex have been shown to be proportional to the energy of the complete graph. The results of splitting the graph and shadow graphs are also presented for the complete graph after removing a vertex.

Keywords: Splitting graph, Shahow graph, Maximum degree energy, Maximum reverse degree energy, Graph energy, Matrices, Eigenvalues

1. Introduction

The study of spectral graph theory explores connections between the algebraic characteristics of the spectra of specific matrices associated with a graph. Various matrices, including the Laplacian matrix, incidence matrix, adjacency matrix, and distance matrix, are connected to a graph.

The idea of graph energy originated in quantum chemistry in 1930 when E. Huckel introduced the chemical applications of graph theory in molecular orbital theory for the $π$ -electron network of conjugated hydrocarbons, leading to the discovery of eigenvalues.

In [1], the distance spectrum and distance energy of some families of graphs are calculated. Hampiholi in [2] presents results for different energies for the shadow graph. In [3], the author defines maximum reverse degree energy, provides some properties, and generates results for this energy for some families. In [4], the maximum degree energy of a graph is defined, and bounds for its results are given. [5] establishes relations between the maximum energy and minimum energy of the shadow and splitting graphs of a graph. In [6], the characteristic polynomial of the minimum degree matrix of graphs obtained by certain graph operations is discussed, along with bounds for the largest minimum degree eigenvalue and minimum degree energy of graphs. [7] provides the energy of a graph class in terms of another graph class after removing a vertex. In [8], the corresponding energy of a given graph $G$ is expressed as a multiple of various graph energies of the regular splitting graph $S (G)$ . Numerous applications of graph theory in computer science and engineering are detailed in [9]. [10] proves that the splitting graph $S^{'} (G)$ is a Zagreb hyper-energetic graph. In [11], various lower and upper bounds for graph energy $E (G)$ are derived. [12] presents bounds and characterizations on the largest eigenvalue of the Sombor matrix $S (G)$ and Sombor energy of graphs. Statistical information on the research of graph energies and their applications is provided in [13]. [14] introduces the Sombor characteristic polynomial and the Sombor energy for some graph classes. In [15], the author establishes the relationship between energy and Sombor energy of the m-splitting graph and m-shadow graph of a k-regular graph. [16] calculates average degree eigenvalues and average degree energy for some families of graphs. In [17], Kousar and Nazeer present results for numerous graph energies of the regular subdivision graph and complete graph. The study of the energies of some classes of non-regular graphs is shown in [18].

This paper is organized as follows: Section 2 presents the results of the maximum degree energy of the splitting graph and shadow graph of a complete graph after removing a vertex. Section 3 discusses the results of the maximum reverse degree energy of the splitting graph and shadow graph of a complete graph after removing a vertex.

2. Preliminaries

Let $G$ be a simple, undirected and finite graph and let its vertex set and edge set be denoted by $V (G) = {v_{1}, v_{2}, v_{3}, \dots, v_{p}}$ and $E (G) = {e_{1}, e_{2}, e_{3}, \dots, e_{q}}$ respectively. The number of edges associated with a vertex $v$ of a graph $G$ is called the degree of vertex $v$ and is denoted by $d_{v}$ or $d (v)$ [19].

Let $λ_{1}, λ_{2}, λ_{3}, \dots, λ_{k}$ be the eigenvalues of $G$ . Gutman in 1978 [20] defined the energy of the graph $G$ as the sum of the absolute values of all eigenvalues of $G$ . Therefore, $E (G) = \sum_{i = 1}^{k} | λ_{i} | .$

The maximum degree matrix $M_{e} (G) = [M_{i j}]$ [4] of $G$ is defined as $[M_{i j}] = {\begin{cases} max (d_{i}, d_{j}), & if v_{i} and v_{j} are adjacent; \\ 0, & otherwise . \end{cases}$

Adiga defined the maximum degree energy $M (G)$ of a simple connected graph $G$ as the sum of the absolute values of eigenvalues of the maximum degree matrix $M_{[i j]}$ of $G$ .

Let $Δ (G)$ denote the maximum degree among the vertices of $G$ . The reverse vertex degree of a vertex $v_{i}$ in $G$ is defined as $c_{v_{i}} = Δ (G) - d (v_{i}) + 1$ where $d (v_{i})$ is the degree of vertex $v_{i}$ . Then the maximum reverse degree matrix is defined as $M_{R} (G) = (r_{i j})$ [3], where $[r_{i j}] = {\begin{cases} max (c_{i}, c_{j}), & if v_{i} and v_{j} areadjacent; \\ 0, & otherwise . \end{cases}$

The maximum reverse degree energy $M_{R} (G)$ of a simple connected graph $G$ is the sum of the absolute values of eigenvalues of the maximum reverse degree matrix $r_{[i j]}$ of $G$ .

3. Maximum Degree Energy of Complete Graph

Theorem 1. For the complete graph $K_{n}$ with $n \geq 4$ , the maximum degree energy $M_{e} (K_{n})$ is given by $M_{e} (K_{n}) = 2 \cdot deg (v)^{2}$ , where $v$ is any vertex.

Proof. The conclusion is derived from the fundamental properties of complete graphs $K_{n}$ , where each vertex shares an edge with every other vertex, resulting in a degree of $(n - 1)$ for each vertex.

Vertices $(n \geq 4)$	Degree $deg (K_{n})$	Maximum Degree Energy $M_{e} (K_{n})$
$n = 4$	$deg (K_{4}) = 3$	$M_{e} (K_{4}) = 18$
$n = 5$	$deg (K_{5}) = 4$	$M_{e} (K_{5}) = 32$
$n = 6$	$deg (K_{6}) = 5$	$M_{e} (K_{6}) = 50$
⋮	⋮	⋮
$n t h$	$deg (K_{n}) = n - 1$	$M_{e} (K_{n}) = 2 \cdot deg (v)^{2}$

Theorem 2. For a complete graph $K_{n}$ with $n \geq 4$ , the maximum degree energy $M_{e} (K_{n})$ satisfies the relation $M_{e} [K_{n}] = {(\frac{deg (v)}{deg (v) - 1})}^{2} M_{e} [K_{n} - v] .$

Proof. Consider a complete graph $K_{n}$ where each vertex shares exactly one edge with every other vertex, resulting in a degree of $(n - 1)$ for each vertex. We will prove the given relation using mathematical induction.

Basic step: Let $n = 5$ , then $\begin{aligned} M_{e} [K_{5}] & = {(\frac{deg (v)}{deg (v) - 1})}^{2} M_{e} [K_{5} - v] . \\ = {(\frac{deg (v)}{deg (v) - 1})}^{2} M_{e} [K_{4}] . \\ = {(\frac{4}{3})}^{2} (18) = 32 = M_{e} [K_{5}] . \end{aligned}$ Hence, the result holds for $n = 5$ .

Induction hypothesis: Assume that the result holds for $n = m$ , i.e., $\begin{aligned} M_{e} [K_{m}] & = {(\frac{deg (v)}{deg (v) - 1})}^{2} M_{e} [K_{m} - v] \\ = {(\frac{(m - 1)}{(m - 2)})}^{2} M_{e} [K_{m - 1}] . \end{aligned}$

Induction step: Now, we prove the result for $n = (m + 1)$ . $\begin{aligned} M_{e} [K_{m + 1}] & = {(\frac{m}{m - 1})}^{2} M_{e} [K_{m}] \\ = {(\frac{m}{m - 1})}^{2} {(\frac{deg (v)}{deg (v) - 1})}^{2} M_{e} [K_{m} - v] \\ = {(\frac{deg (v)}{deg (v) - 1})}^{2} {(\frac{m}{m - 1})}^{2} M_{e} [K_{m - 1}] \\ = {(\frac{m - 1}{m - 2})}^{2} {(\frac{m}{m - 1})}^{2} M_{e} [K_{m - 1}] \\ = {(\frac{m}{m - 1})}^{2} {(\frac{m - 1}{m - 2})}^{2} M_{e} [K_{m - 1}] \end{aligned}$ $\begin{matrix} (1) & = {(\frac{deg (v)}{deg (v) - 1})}^{2} M_{e} [K_{m}] . \end{matrix}$ Eq. (1) demonstrates that the result holds for $n = (m + 1)$ .

Since mathematical induction performs the basis and induction steps, the result holds for all $n \geq 5$ .

Vertices $(n \geq 5)$	$M_{e} (K_{n})$	$M_{e} (K_{n} - v)$
$n = 5$	$M_{e} (K_{5}) = 32$	$M_{e} (K_{5} - v) = 18$
$n = 6$	$M_{e} (K_{6}) = 50$	$M_{e} (K_{6} - v) = 32$
$n = 7$	$M_{e} (K_{7}) = 72$	$M_{e} (K_{7} - v) = 50$
$n = 8$	$M_{e} (K_{8}) = 98$	$M_{e} (K_{8} - v) = 72$
⋮	⋮	⋮

Corollary 1. For $n \geq 5$ , the difference in energy between a complete graph $K_{n}$ and its vertex deletion $K_{n} - v$ is given by $M_{e} (K_{n}) - M_{e} (K_{n} - v) = 4 n + 14.$

Vertices $(n \geq 5)$	$M_{e} (K_{n})$	$M_{e} (K_{n} - v)$	Difference
$n = 5$	$M_{e} (K_{5}) = 32$	$M_{e} (K_{5} - v) = 18$	$14$
$n = 6$	$M_{e} (K_{6}) = 50$	$M_{e} (K_{6} - v) = 32$	$18$
$n = 7$	$M_{e} (K_{7}) = 72$	$M_{e} (K_{7} - v) = 50$	$22$
$n = 8$	$M_{e} (K_{8}) = 98$	$M_{e} (K_{8} - v) = 72$	$26$
⋮	⋮	⋮	⋮

4. Maximum Degree Energy of Splitting Graph

Theorem 3. For the splitting graph $Sp (K_{n} - v)$ , where $n \geq 4$ , we have $M_{e} (Sp (K_{n} - v)) = 2 \sqrt{5} M_{e} (K_{n - 1}) .$

Proof. Consider a complete graph $K_{6}$ , where $K_{6} - v$ denotes the deletion of a vertex from $K_{6}$ and $Sp (K_{6} - v)$ represents the splitting graph. To demonstrate that $M_{e} (Sp (K_{6} - v)) = 2 \sqrt{5} M_{e} (K_{6 - 1}),$ we calculate the energy of $Sp (K_{6} - v)$ .

$M_{e} (Sp (K_{6} - v)) = [\begin{matrix} 0 & 8 & 8 & 8 & 8 & 0 & 8 & 8 & 8 & 8 \\ 8 & 0 & 8 & 8 & 8 & 8 & 0 & 8 & 8 & 8 \\ 8 & 8 & 0 & 8 & 8 & 8 & 8 & 0 & 8 & 8 \\ 8 & 8 & 8 & 0 & 8 & 8 & 8 & 8 & 0 & 8 \\ 8 & 8 & 8 & 8 & 0 & 8 & 8 & 8 & 8 & 0 \\ 0 & 8 & 8 & 8 & 8 & 0 & 0 & 0 & 0 & 0 \\ 8 & 0 & 8 & 8 & 8 & 0 & 0 & 0 & 0 & 0 \\ 8 & 8 & 0 & 8 & 8 & 0 & 0 & 0 & 0 & 0 \\ 8 & 8 & 8 & 0 & 8 & 0 & 0 & 0 & 0 & 0 \\ 8 & 8 & 8 & 8 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$

The characteristic polynomial of $M_{e} (Sp (K_{6} - v))$ is: $\begin{aligned} λ^{10} - 1920 λ^{8} - 40960 λ^{7} - 20480 λ^{6} + 5111808 λ^{5} + 13107200 λ^{4} \\ - 293601280 λ^{3} - 25168240 λ^{2} + 8053063680 λ - 17179869184 = 0. \end{aligned}$

The eigenvalues are: $λ_{1} = 51.7771, λ_{2} = λ_{3} = λ_{4} = λ_{5} = 4.9443, λ_{6} = λ_{7} = λ_{8} = λ_{9} = - 12.9443, λ_{10} = - 19.7771 .$

Thus, the spectral matrix $Spec (M_{e} (Sp (K_{6} - v)))$ is: $(\begin{matrix} - 19.7771 & - 12.9443 & 4.9443 & 51.7771 \\ 1 & 4 & 4 & 1 \end{matrix})$

Hence, $M_{e} (Sp (K_{6} - v)) = 143.1084 = 2 \sqrt{5} M_{e} (Sp (K_{5})) = 2 \sqrt{5} M_{e} (K_{6} - v)$ .

$M_{e} (K_{n})$	$M_{e} (Sp (K_{n} - v))$	$M_{e} (K_{n} - v) = 2 \sqrt{5} M_{e} (K_{n - 1})$
$M_{e} (K_{6}) = 50$	$M_{e} (Sp (K_{6} - v)) = 143.1084$	$M_{e} (Sp (K_{6} - v)) = 2 \sqrt{5} (32)$
$M_{e} (K_{7}) = 72$	$M_{e} (Sp (K_{7} - v)) = 223.6068$	$M_{e} (Sp (K_{7} - v)) = 2 \sqrt{5} (50)$
$M_{e} (K_{8}) = 98$	$M_{e} (Sp (K_{8} - v)) = 321.9938$	$M_{e} (Sp (K_{8} - v)) = 2 \sqrt{5} (72)$
$M_{e} (K_{9}) = 128$	$M_{e} (Sp (K_{9} - v)) = 438.2698$	$M_{e} (Sp (K_{9} - v)) = 2 \sqrt{5} (98)$
⋮	⋮	⋮

5. Maximum Degree Energy of Shadow Graph

Theorem 4. For a complete graph $K_{n}$ with $n \geq 6$ , the maximum degree energy of its shadow graph is given by $M_{e} (D (K_{n} - v)) = 4 M_{e} (K_{n - 1}) .$

Proof. Consider a complete graph $K_{6}$ . Let $K_{6} - v$ denote the deletion of a vertex from $K_{6}$ , and let $D (K_{6} - v)$ represent its shadow graph. We aim to demonstrate that $M_{e} (D (K_{6} - v)) = 4 M_{e} (K_{6} - v) .$

The maximum degree energy matrix $M_{e} (D (K_{6} - v))$ for $K_{6} - v$ is as follows: $[\begin{matrix} 0 & 8 & 8 & 8 & 8 & 0 & 8 & 8 & 8 & 8 \\ 8 & 0 & 8 & 8 & 8 & 8 & 0 & 8 & 8 & 8 \\ 8 & 8 & 0 & 8 & 8 & 8 & 8 & 0 & 8 & 8 \\ 8 & 8 & 8 & 0 & 8 & 8 & 8 & 8 & 0 & 8 \\ 8 & 8 & 8 & 8 & 0 & 8 & 8 & 8 & 8 & 0 \\ 0 & 8 & 8 & 8 & 8 & 0 & 8 & 8 & 8 & 8 \\ 8 & 0 & 8 & 8 & 8 & 8 & 0 & 8 & 8 & 8 \\ 8 & 8 & 0 & 8 & 8 & 8 & 8 & 0 & 8 & 8 \\ 8 & 8 & 8 & 0 & 8 & 8 & 8 & 8 & 0 & 8 \\ 8 & 8 & 8 & 8 & 0 & 8 & 8 & 8 & 8 & 0 \end{matrix}]$

The characteristic polynomial of $M_{e} (D (K_{6} - v))$ is $λ^{10} - 2560 λ^{8} - 8190 λ^{7} - 983040 λ^{6} - 4194304 λ^{5} = 0.$

Its eigenvalues are $λ_{1} = 64$ , $λ_{2} = 8.5850 e^{- 15}$ , $λ_{3} = 5.3785 e^{- 15}$ , $λ_{4} = 3.8317 e^{- 15}$ , $λ_{5} = 1.1914 e^{- 15}$ , $λ_{6} = - 4.3013 e^{- 15}$ , $λ_{7} = λ_{8} = λ_{9} = λ_{10} = - 16$ . Thus, $Spec (M_{e} (D (K_{6} - v))) = (\begin{matrix} - 16 & - 4.3013 e^{- 15} & 1.1914 e^{- 15} & 3.8317 e^{- 15} & 5.3785 e^{- 15} & 8.5850 e^{- 15} & 64 \\ 4 & 1 & 1 & 1 & 1 & 1 & 1 \end{matrix})$

Therefore, $M_{e} (D (K_{6} - v)) = 128 = 4 M_{e} (K_{5}) = 4 M_{e} (K_{6} - v)$ .

The Table below demonstrates the relationship between the maximum degree energies of the complete graph and its shadow graph after vertex deletion:

$M_{e} (K_{n})$	$M_{e} (D (K_{n} - v))$	$M_{e} (D (K_{n} - v)) = 4 M_{e} (K_{n - 1})$
$M_{e} (K_{6}) = 50$	$M_{e} (D (K_{6} - v)) = 128$	$M_{e} (D (K_{6} - v)) = 4 (32)$
$M_{e} (K_{7}) = 72$	$M_{e} (D (K_{7} - v)) = 200$	$M_{e} (D (K_{7} - v)) = 4 (50)$
$M_{e} (K_{8}) = 98$	$M_{e} (D (K_{8} - v)) = 288$	$M_{e} (D (K_{8} - v)) = 4 (72)$
$M_{e} (K_{9}) = 128$	$M_{e} (D (K_{9} - v)) = 392$	$M_{e} (D (K_{9} - v)) = 4 (98)$
⋮	⋮	⋮

6. Maximum Reverse Degree Energy of Complete Graph

Theorem 5. For the complete graph $K_{n}$ with $n \geq 4$ , we have $M_{R} [K_{n}] = 2 (n - 1)$ , where $v$ is any vertex.

Proof. The conclusion follows from the definition of a complete graph $K_{n}$ , where each vertex has the same degree $(n - 1)$ .

Vertices $(n \geq 4)$	$deg (K_{n})$	$M_{R} (K_{n})$
$n = 4$	$deg (K_{4}) = 3$	$M_{R} (K_{4}) = 6$
$n = 5$	$deg (K_{5}) = 4$	$M_{R} (K_{5}) = 8$
$n = 6$	$deg (K_{6}) = 5$	$M_{R} (K_{6}) = 10$
⋮	⋮	⋮
$n t h$	$deg (K_{n}) = n - 1$	$M_{R} (K_{n}) = 2 (n - 1)$

7. Maximum Reverse Degree Energy of Complete Graph

Theorem 6. For a complete graph $K_{n}$ where $n \geq 4$ , $M_{R} [K_{n}] = \frac{deg (v)}{deg (v) - 1} M_{R} [K_{n} - v] .$

Proof. Consider a complete graph $K_{n}$ where each vertex has degree $(n - 1)$ , and exactly one edge is shared by every pair of vertices. We utilize the principle of mathematical induction to prove this conclusion.

Base step: Let $n = 5$ . Then, $\begin{aligned} M_{R} [K_{5}] & = \frac{deg (v)}{deg (v) - 1} M_{R} [K_{5} - v] \\ = \frac{deg (v)}{deg (v) - 1} M_{R} [K_{5} - v] \\ = \frac{deg (v)}{deg (v) - 1} M_{R} [K_{4}] \\ = \frac{4}{3} (6) = 8 = M_{R} [K_{5}] . \end{aligned}$ Hence, the result is true for $n = 5$ .

Induction hypothesis: Assume that the result holds for $n = m$ . Then, $\begin{aligned} M_{R} [K_{m}] & = \frac{deg (v)}{deg (v) - 1} M_{R} [K_{m} - v] \\ = \frac{deg (v)}{deg (v) - 1} M_{R} [K_{m - 1}] \\ = \frac{(m - 1)}{(m - 2)} M_{R} [K_{m - 1}] . \end{aligned}$

Induction step: Now, we prove the result for $n = (m + 1)$ . That is, $M_{R} [K_{m + 1}] = \frac{deg (v)}{deg (v) - 1} M_{R} [K_{m + 1} - v] .$ Consider $\begin{aligned} M_{R} [K_{m + 1}] & = \frac{m}{m - 1} M_{R} [K_{m}] \\ = (\frac{m}{m - 1}) \frac{deg (v)}{deg (v) - 1} M_{R} [K_{m} - v] \\ = \frac{deg (v)}{deg (v) - 1} (\frac{m}{m - 1}) M_{R} [K_{m - 1}] \\ = (\frac{m - 1}{m - 2}) (\frac{m}{m - 1}) M_{R} [K_{m - 1}] \\ = (\frac{m}{m - 1}) (\frac{m - 1}{m - 2}) M_{R} [K_{m - 1}] \end{aligned}$ $\begin{matrix} (2) & = \frac{deg (v)}{deg (v) - 1} M_{R} [K_{m}] . \end{matrix}$ The Eq. (2) demonstrates that the result is accurate for $n = (m + 1)$ .

Eq. (2) is true because mathematical induction performs the basis and induction stages. Hence, the result holds for all $n \geq 5$ .

Vertices $(n \geq 5)$	$M_{R} (K_{n})$	$M_{R} (K_{n} - v)$
$n = 5$	$M_{R} (K_{5}) = 8$	$M_{R} (K_{5} - v) = 6$
$n = 6$	$M_{R} (K_{6}) = 10$	$M_{R} (K_{6} - v) = 8$
$n = 7$	$M_{R} (K_{7}) = 12$	$M_{R} (K_{7} - v) = 10$
$n = 8$	$M_{R} (K_{8}) = 14$	$M_{R} (K_{8} - v) = 12$
⋮	⋮	⋮

Corollary 2. The energy difference between a complete graph $K_{n}$ and its vertex deletion $K_{n} - v$ is equal to $2$ for $n \geq 5$ : $M_{R} (K_{n}) - M_{R} (K_{n} - v) = 2.$

Vertices $(n \geq 5)$	$M_{R} (K_{n})$	$M_{R} (K_{n} - v)$	Difference
$n = 5$	$M_{R} (K_{5}) = 8$	$M_{R} (K_{5} - v) = 6$	$2$
$n = 6$	$M_{R} (K_{6}) = 10$	$M_{R} (K_{6} - v) = 8$	$2$
$n = 7$	$M_{R} (K_{7}) = 12$	$M_{R} (K_{7} - v) = 10$	$2$
$n = 8$	$M_{R} (K_{8}) = 14$	$M_{R} (K_{8} - v) = 12$	$2$
⋮	⋮	⋮	⋮

8. Maximum Reverse Degree Energy of Splitting Graph

Theorem 7. For a complete graph $K_{n}$ where $n \geq 4$ , $M_{R} (Sp (K_{n} - v)) = \sqrt{4 n^{2} + 1} \cdot M_{R} (K_{n - 1}) .$

Proof. Consider a complete graph $K_{6}$ , where $K_{6} - v$ is the deletion of a vertex from $K_{6}$ , and $Sp (K_{6} - v)$ is a splitting graph. We aim to prove that $M_{R} (Sp (K_{6} - v)) = \sqrt{4 n^{2} + 1} \cdot M_{R} (K_{6 - 1}) .$

$M_{R} (Sp (K_{6} - v)) = [\begin{matrix} 0 & 1 & 1 & 1 & 1 & 0 & 5 & 5 & 5 & 5 \\ 1 & 0 & 1 & 1 & 1 & 5 & 0 & 5 & 5 & 5 \\ 1 & 1 & 0 & 1 & 1 & 5 & 5 & 0 & 5 & 5 \\ 1 & 1 & 1 & 0 & 1 & 5 & 5 & 5 & 0 & 5 \\ 1 & 1 & 1 & 1 & 0 & 5 & 5 & 5 & 5 & 0 \\ 0 & 5 & 5 & 5 & 5 & 0 & 0 & 0 & 0 & 0 \\ 5 & 0 & 5 & 5 & 5 & 0 & 0 & 0 & 0 & 0 \\ 5 & 5 & 0 & 5 & 5 & 0 & 0 & 0 & 0 & 0 \\ 5 & 5 & 5 & 0 & 5 & 0 & 0 & 0 & 0 & 0 \\ 5 & 5 & 5 & 5 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$

The characteristic polynomial of $M_{R} (Sp (K_{6} - v))$ is $\begin{aligned} λ^{10} - 510 λ^{8} - 1520 λ^{7} + 42235 λ^{6} + 111996 λ^{5} - 1468750 λ^{4} \\ - 2787500 λ^{3} + 24140625 λ^{2} + 23437500 λ - 15620000 = 0. \end{aligned}$ Its eigenvalues are $\begin{aligned} λ_{1} & = 22.0998, \\ λ_{2} & = λ_{3} = λ_{4} = λ_{5} = 4.5249, \\ λ_{6} & = λ_{7} = λ_{8} = λ_{9} = - 5.5249, \\ λ_{10} & = - 18.0998 . \end{aligned}$

$Spec (M_{e} (Sp (K_{6} - v))) = (\begin{matrix} - 18.0998 & - 5.5249 & 4.5249 & 22.0998 \\ 1 & 4 & 4 & 1 \end{matrix})$

Thus, $M_{R} (Sp (K_{6} - v)) = 80.3990 = \sqrt{4 n^{2} + 1} \cdot M_{R} (K_{5}) = \sqrt{4 n^{2} + 1} \cdot M_{R} (K_{6} - v) .$

$M_{R} (K_{n})$	$M_{R} (Sp (K_{n} - v))$	$M_{R} (K_{n} - v) = \sqrt{4 n^{2} + 1} \cdot M_{R} (K_{n - 1})$
$M_{R} (K_{6}) = 10$	$M_{R} (Sp (K_{6} - v)) = 80.3990$	$M_{R} (Sp (K_{6} - v)) = \sqrt{4 n^{2} + 1} \cdot (8)$
$M_{R} (K_{7}) = 12$	$M_{R} (Sp (K_{7} - v)) = 120.4159$	$M_{R} (Sp (K_{7} - v)) = \sqrt{4 n^{2} + 1} \cdot (10)$
$M_{R} (K_{8}) = 14$	$M_{R} (Sp (K_{8} - v)) = 168.4280$	$M_{R} (Sp (K_{8} - v)) = \sqrt{4 n^{2} + 1} \cdot (12)$
$M_{R} (K_{9}) = 16$	$M_{R} (Sp (K_{9} - v)) = 224.4371$	$M_{R} (Sp (K_{9} - v)) = \sqrt{4 n^{2} + 1} \cdot (14)$
⋮	⋮	⋮

9. Maximum Reverse Degree Energy of Shadow Graph

Theorem 8. For a complete graph $K_{n}$ where $n \geq 4$ , $M_{R} (D (K_{n} - v)) = 2 n \cdot M_{R} (K_{n - 1}) .$

Proof. Consider a complete graph $K_{6}$ , where $K_{6} - v$ is the deletion of a vertex from $K_{6}$ , and $D (K_{6} - v)$ is a shadow graph. We aim to prove that $M_{R} (D (K_{6} - v)) = 2 n \cdot M_{R} (K_{6} - v) .$

$M_{R} (D (K_{6} - v)) = [\begin{matrix} 0 & 1 & 1 & 1 & 1 & 0 & 5 & 5 & 5 & 5 \\ 1 & 0 & 1 & 1 & 1 & 5 & 0 & 5 & 5 & 5 \\ 1 & 1 & 0 & 1 & 1 & 5 & 5 & 0 & 5 & 5 \\ 1 & 1 & 1 & 0 & 1 & 5 & 5 & 5 & 0 & 5 \\ 1 & 1 & 1 & 1 & 0 & 5 & 5 & 5 & 5 & 0 \\ 0 & 5 & 5 & 5 & 5 & 0 & 1 & 1 & 1 & 1 \\ 5 & 0 & 5 & 5 & 5 & 1 & 0 & 1 & 1 & 1 \\ 5 & 5 & 0 & 5 & 5 & 1 & 1 & 0 & 1 & 1 \\ 5 & 5 & 5 & 0 & 5 & 1 & 1 & 1 & 0 & 1 \\ 5 & 5 & 5 & 5 & 0 & 1 & 1 & 1 & 1 & 0 \end{matrix}]$

The characteristic polynomial of $M_{R} (D (K_{6} - v))$ is $\begin{aligned} λ^{10} - 520 λ^{8} - 3040 λ^{7} + 34320 λ^{6} + 203392 λ^{5} - 1036800 λ^{4} \\ - 4792320 λ^{3} + 17141760 λ^{2} + 39813120 λ - 127401984 = 0. \end{aligned}$ Its eigenvalues are $λ_{1} = 24, λ_{2} = λ_{3} = λ_{4} = λ_{5} = 4, λ_{6} = λ_{7} = λ_{8} = λ_{9} = - 6, λ_{10} = - 16.$

$Spec (M_{R} (D (K_{6} - v))) = (\begin{matrix} - 16 & - 6 & 4 & 24 \\ 1 & 4 & 4 & 1 \end{matrix})$

Thus, $M_{R} (D (K_{6} - v)) = 80 = 2 n \cdot M_{R} (K_{5}) = 2 n \cdot M_{R} (K_{6} - v) .$

$M_{R} (K_{n})$	$M_{R} (D (K_{n} - v))$	$M_{R} (D (K_{n} - v)) = 2 n \cdot M_{R} (K_{n - 1})$
$M_{R} (K_{6}) = 10$	$M_{R} (D (K_{6} - v)) = 80$	$M_{R} (D (K_{6} - v)) = 2 n \cdot (8)$
$M_{R} (K_{7}) = 12$	$M_{R} (D (K_{7} - v)) = 120$	$M_{R} (D (K_{7} - v)) = 2 n \cdot (10)$
$M_{R} (K_{8}) = 14$	$M_{R} (D (K_{8} - v)) = 168$	$M_{R} (D (K_{8} - v)) = 2 n \cdot (12)$
$M_{R} (K_{9}) = 16$	$M_{R} (D (K_{9} - v)) = 224$	$M_{R} (D (K_{9} - v)) = 2 n \cdot (14)$
⋮	⋮	⋮

10. Conclusion

In this study, we have examined the family of complete graphs and their associated splitting and shadow graphs to investigate their energies after the deletion of a vertex. Specifically, we have established the relationship between the maximum degree energy and the maximum reverse degree energy of complete graphs and their splitting and shadow graphs for vertex deletions.

References:

Indulal, G., Gutmanb, I. and Vijayakumar, A., 2008. On distance energy of graphs. MATCH Communications in Mathematical and in Computer Chemistry, 60, pp.461-472.
Hampiholi, P., Joshi, A.S., Deshpande, V.V. and Adhyapak, P., 2020. On Various Graph Energies Of Shadow Graph. Asian Journal of Mathematics and Computer Research, pp.27-36.
Jayanna, G.K., 2023. On maximum reverse degree energy of a graph and its chemical applicability. Bulletin of International Mathematical Virtual Institute, 13(1), pp.83-95.
Adiga, C. and Smitha, M., 2009. On maximum degree energy of a graph. International Journal of Contemporary Mathematical Sciences, 4(8), pp.385-396.
Rao, K. S., Saravanan, K., Prakasha, K. N. and Cangul, I. N., 2022. Maximum and minimum degree energies of p-splitting and p-shadow graphs. TWMS Journal of Applied and Engineering Mathematics, 12(1), pp.1-10.
Basavanagoud, B. and Jakkannavar, P., 2019. Minimum degree energy of graphs. Electronic Journal of Mathematical Analysis and Applications, 7(2), pp.230-243.
Sangeetha, G. and Kavitha, J., 2022. Results on Graph Energy. Journal of Physics: Conference Series, 2332(1), p.012008. IOP Publishing.
Chu, Z. Q., Nazeer, S., Zia, T. J., Ahmed, I. and Shahid, S., 2019. Some new results on various graph energies of the splitting graph. Journal of Chemistry, 2019, pp.1-12.
Singh, R. P., 2014. Application of graph theory in computer science and engineering. International Journal of Computer Applications, 104(1), pp.10-13.
Sheikholeslami, S. M., Jahanbani, A. and Khoeilar, R., 2021. New Results on Zagreb Energy of Graphs. Mathematical Problems in Engineering, 2021, pp.1-6.
Filipovski, S. and Jajcay, R., 2021. Bounds for the energy of graphs. Mathematics, 9(14), p.1687.
Gowtham, K. J. and Narasimha, S. N., 2021. On Sombor energy of graphs. Nanosystems: Physics, Chemistry, Mathematics , 12(4), pp.411-417.
Gutman, I. and Furtula, B., 2019. Graph energies and their applications. Bulletin (Académie serbe des sciences et des arts. Classe des sciences mathématiques et naturelles. Sciences Mathématiques, (44), pp.29-45.
Ghanbari, N., 2022. On the Sombor characteristic polynomial and Sombor energy of a graph. Computational and Applied Mathematics, 41(6), p.242.
Singh, R. and Patekar, S.C., 2022. On the Sombor index and Sombor energy of m-splitting graph and m-shadow graph of regular graphs. arXiv preprint arXiv:2205.09480.
Patekar, S. C., Barde, S. A. and Shikare, M. M., 2019. On the average degree eigenvalues and average degree energy of graphs. Journal of Mathematics and Computer Science, 9(1), pp.46-59.
Kousar, I., Nazeer, S., Mahboob, A., Shahid, S. and Lv, Y. P., 2021. Numerous graph energies of regular subdivision graph and complete graph. AIMS Mathematics, 6(8), pp.8466-8476. AIMS Press.
Indulal, G. and Vijayakumar, A., 2007. Energies of some non-regular graphs. Springer.
Wilson, R. J., 1979. Introduction to Graph Theory. Pearson Education India.
Gutman, I., 2001. The energy of a graph: old and new results. In Algebraic Combinatorics and Applications: Proceedings of the Euroconference, Algebraic Combinatorics and Applications (ALCOMA), held in Gö $β$ weinstein, Germany, September 12–19, 1999 (pp. 196-211). Berlin, Heidelberg: Springer Berlin Heidelberg.

[1] Indulal, G., Gutmanb, I. and Vijayakumar, A., 2008. On distance energy of graphs. MATCH Communications in Mathematical and in Computer Chemistry, 60, pp.461-472.

[2] Hampiholi, P., Joshi, A.S., Deshpande, V.V. and Adhyapak, P., 2020. On Various Graph Energies Of Shadow Graph. Asian Journal of Mathematics and Computer Research, pp.27-36.

[3] Jayanna, G.K., 2023. On maximum reverse degree energy of a graph and its chemical applicability. Bulletin of International Mathematical Virtual Institute, 13(1), pp.83-95.

[4] Adiga, C. and Smitha, M., 2009. On maximum degree energy of a graph. International Journal of Contemporary Mathematical Sciences, 4(8), pp.385-396.

[5] Rao, K. S., Saravanan, K., Prakasha, K. N. and Cangul, I. N., 2022. Maximum and minimum degree energies of p-splitting and p-shadow graphs. TWMS Journal of Applied and Engineering Mathematics, 12(1), pp.1-10.

[6] Basavanagoud, B. and Jakkannavar, P., 2019. Minimum degree energy of graphs. Electronic Journal of Mathematical Analysis and Applications, 7(2), pp.230-243.

[7] Sangeetha, G. and Kavitha, J., 2022. Results on Graph Energy. Journal of Physics: Conference Series, 2332(1), p.012008. IOP Publishing.

[8] Chu, Z. Q., Nazeer, S., Zia, T. J., Ahmed, I. and Shahid, S., 2019. Some new results on various graph energies of the splitting graph. Journal of Chemistry, 2019, pp.1-12.

[9] Singh, R. P., 2014. Application of graph theory in computer science and engineering. International Journal of Computer Applications, 104(1), pp.10-13.

[10] Sheikholeslami, S. M., Jahanbani, A. and Khoeilar, R., 2021. New Results on Zagreb Energy of Graphs. Mathematical Problems in Engineering, 2021, pp.1-6.

[11] Filipovski, S. and Jajcay, R., 2021. Bounds for the energy of graphs. Mathematics, 9(14), p.1687.

[12] Gowtham, K. J. and Narasimha, S. N., 2021. On Sombor energy of graphs. Nanosystems: Physics, Chemistry, Mathematics , 12(4), pp.411-417.

[13] Gutman, I. and Furtula, B., 2019. Graph energies and their applications. Bulletin (Académie serbe des sciences et des arts. Classe des sciences mathématiques et naturelles. Sciences Mathématiques, (44), pp.29-45.

[14] Ghanbari, N., 2022. On the Sombor characteristic polynomial and Sombor energy of a graph. Computational and Applied Mathematics, 41(6), p.242.

[15] Singh, R. and Patekar, S.C., 2022. On the Sombor index and Sombor energy of m-splitting graph and m-shadow graph of regular graphs. arXiv preprint arXiv:2205.09480.

[16] Patekar, S. C., Barde, S. A. and Shikare, M. M., 2019. On the average degree eigenvalues and average degree energy of graphs. Journal of Mathematics and Computer Science, 9(1), pp.46-59.

[17] Kousar, I., Nazeer, S., Mahboob, A., Shahid, S. and Lv, Y. P., 2021. Numerous graph energies of regular subdivision graph and complete graph. AIMS Mathematics, 6(8), pp.8466-8476. AIMS Press.

[18] Indulal, G. and Vijayakumar, A., 2007. Energies of some non-regular graphs. Springer.

[19] Wilson, R. J., 1979. Introduction to Graph Theory. Pearson Education India.

[20] Gutman, I., 2001. The energy of a graph: old and new results. In Algebraic Combinatorics and Applications: Proceedings of the Euroconference, Algebraic Combinatorics and Applications (ALCOMA), held in Gö $β$ weinstein, Germany, September 12–19, 1999 (pp. 196-211). Berlin, Heidelberg: Springer Berlin Heidelberg.

Contents

Utilitas Mathematica

On Maximum Degree and Maximum Reverse Degree Energies of Splitting and Shadow graph of Complete graph

Abstract

1. Introduction

2. Preliminaries

3. Maximum Degree Energy of Complete Graph

4. Maximum Degree Energy of Splitting Graph

5. Maximum Degree Energy of Shadow Graph

6. Maximum Reverse Degree Energy of Complete Graph

7. Maximum Reverse Degree Energy of Complete Graph

8. Maximum Reverse Degree Energy of Splitting Graph

9. Maximum Reverse Degree Energy of Shadow Graph

10. Conclusion

References:

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