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Journal of Chemistry
Volume 2019, Article ID 7214047, 12 pages
https://doi.org/10.1155/2019/7214047
Research Article

Some New Results on Various Graph Energies of the Splitting Graph

1Department of Mathematics and Physics, Anhui Xinhua University, 230088 Hefei, China
2Lahore College for Women University, Lahore, Pakistan
3COMSATS University of Islamabad, Lahore Campus, Lahore 54000, Pakistan

Correspondence should be addressed to Saima Nazeer; moc.oohay@321reezanamias

Received 20 July 2019; Accepted 1 October 2019; Published 19 November 2019

Guest Editor: Shaohui Wang

Copyright © 2019 Zheng-Qing Chu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The energy of a simple connected graph G is equal to the sum of the absolute value of eigenvalues of the graph G where the eigenvalue of a graph G is the eigenvalue of its adjacency matrix . Ultimately, scores of various graph energies have been originated. It has been shown in this paper that the different graph energies of the regular splitting graph is a multiple of corresponding energy of a given graph G.

1. Introduction

Let G be a simple, finite, and undirected graph and its vertex set and edge set are denoted by and , respectively. Number of edges finishing at a vertex of a graph is named as degree of vertex and is denoted by or .

The adjacency matrix of G, denoted by , is a square matrix such that is equal to unity if and is equal to zero otherwise. The eigenvalues of the adjacency matrix are known as the eigenvalues of the graph G. Collection of eigenvalues of the graph G together with their multiplicities is called spectrum of the graph G.

Let be eigenvalues of G and are assumed in nonincreasing order; then, Ivan Gutman in 1978 [1] defined the energy of the graph G as the sum of the absolute values of all eigenvalues of the graph G:

The inspiration of description energy of graph happened from quantum Chemistry. During 1930s, E. Hückel presented chemical applications of graph theory in his molecular orbital theory where eigenvalues of graphs take place. In quantum chemistry, the skeleton of nonsaturated hydrocarbon is represented by a graph. The energy levels of electrons in such a molecule are eigenvalues of graph. The strength of particles is closely identified with the spectrum of its graph. The carbon atoms and chemical bond between them in a hydrocarbon system denote vertices and edges, respectively, in a molecular graph. A lot of work has been done on graph theory, special graph labeling [210], chemical graph theory and graph energies. In the thesis of Siraj [11], certain elementary results on the energy of the graph are also described.

The present work is considered to relate several energies of a graph to bigger graph acquired from the given graph with the help of some graph operations, namely, the splitting graph which is defined in [12]. For a graph G, the splitting graph is obtained by taking a new vertex corresponding to each vertex of the graph G and then join to all vertices of G adjacent to . In [13], it has been proven that

Let and be two matrices of order and , respectively. Then, their tensor product, is obtained from A when every element is replaced by the block and is of order .

Proposition 1 (see [14]). Let and . Also, let α be an eigenvalue of the matrix A with corresponding eigenvector y and β be an eigenvalue of the matrix B with corresponding eigenvector z. Then, is an eigenvalue of with corresponding eigenvector .

In recent times, comparable energies are being considered, based on eigenvalues of a variety of other graph matrices. Numerous matrices can be related to a graph, and their spectrums provide certain helpful information about the graph [1518].

2. Maximum Degree Energy

The maximum degree energy of a simple connected graph G in [19] is defined as the sum of the absolute values of eigenvalues of the maximum degree matrix of a graph G. Then, wherewhere and are the degrees of vertices and , respectively.

Theorem 1. For a graph G,

Proof. Let G be a graph with vertices . Then the maximum degree matrix iswhere and and are the degrees of vertices and , respectively, for and
Let be the vertices corresponding to which are added in G to obtain such that for Then, the maximum degree matrix of is denoted by and can be written as a block matrix:That isHere, the maximum degree spectrum of iswhere for are the eigenvalues of and are the eigenvalues of .
Here,which completes the proof.

3. Minimum Degree Energy

In [20], the minimum degree energy of a simple connected graph G is defined as the sum of the absolute values of eigenvalues of minimum degree matrix of a graph G. Here, wherewhere and are the degrees of vertices and , respectively.

Theorem 2. For a graph G,

Proof. Let G be a graph with vertices . Then, the minimum degree matrix iswhere and and are the degrees of vertices and , respectively, for and Let be the vertices corresponding to which are added in G to obtain such that for . Then, the minimum degree matrix of splitting graph of G, denoted by , can be defined as a block matrix as follows:That isHere, the minimum degree spectrum of iswhere for are the eigenvalues of m(G) and are the eigenvalues of .
Here,which is the required result.

4. Randić Energy

The randić energy of a simple connected graph G in [21] is the sum of the absolute values of eigenvalues of the randić matrix . Here, where

Here, and are the degrees of vertices and , respectively.

Theorem 3. For a graph G,

Proof. Let be vertices of a graph G. Then, the randić matrix of G is denoted by and is given asLet be the vertices corresponding to which are added in G to obtain such that for Then, the randić matrix of is denoted by and can be written as a block matrix as follows: