The Energy and Operations of Graphs

Haicheng Ma,Xiaohua Liu

doi:10.4236/apm.2017.76021

Abstract

Let G be a finite and undirected simple graph on n vertices, A(G) is the adjacency matrix of G, λ1,λ2,...,λn are eigenvalues of A(G), then the energy of G is . In this paper, we determine the energy of graphs obtained from a graph by other unary operations, or graphs obtained from two graphs by other binary operations. In terms of binary operation, we prove that the energy of product graphs is equal to the product of the energy of graphs G1 and G2, and give the computational formulas of the energy of Corona graph , join graph of two regular graphs G and H, respectively. In terms of unary operation, we give the computational formulas of the energy of the duplication graph DmG, the line graph L(G), the subdivision graph S(G), and the total graph T(G) of a regular graph G, respectively. In particularly, we obtained a lot of graphs pair of equienergetic.

Highlights

Let G be a finite and undirected simple graph, with vertex set V (G) and edge set E (G)
We determine the energy of graphs obtained from a graph by other unary operations, or graphs obtained from two graphs by other binary operations
In terms of binary operation, we prove that the energy of product graphs G1 × G2 is equal to the product of the energy of graphs G1 and G2, and give the computational formulas of the energy of Corona graph G H, join graph give the computational formulas of ε (G∇H) of two regular graphs G and H, respectively

Summary

Introduction

Let G be a finite and undirected simple graph, with vertex set V (G) and edge set E (G). The characteristic polynomial of the adjacency matrix, i.e., det ( xIn − A(G)) , where In is the unit matrix of order n, is said to be the characteristic polynomial of the graph G and will be denoted by φ (G, x). The energy of G has been extensively studied, for details, we refer to [7] [8] [9]. In terms of unary operation, we give the computational formulas of the energy of the duplication graph DmG , the line graph L (G) , the subdivision graph S (G) , and the total graph T (G) of a regular graph G, respectively. Let G and H be two vertex disjoint graphs, G ∪ H denotes the union graph of G and H. For more notation and terminology, we refer the readers to standard textbooks [10]

The Binary Operations of Graphs

The Unary Operations of Graphs

Conclusion