Relation Between The Energy And The Edge Dominating Number Of A Graph

Let G be a simple graph with vertex set V(G)and edge set E(G).An edge domi?nating set of G is a subset F of E(G)such that every edge not in F is adjacent to at least one member of F.The edge domination number ?(G)of G is the number of edges in a smallest edge dominating set of G.The energy ?(G)of G is the sum of the absolute values of all eigenvalues of G.We know that an algebraic invariant of graphs-the energy of graphs occupies an important position in graph theory,and it is widely used in physics and chemistry.Gutman extended the concept of energy to all simple graphs,and he de-fined the energy of simple graph G is ?(G)=(?),for ?1,…,?n,are the eigenvalues ofG.Obviously,if we can compute the eigenvalues of a graph,we can immediately know its energy.It is very difficult to compute the eigenvalues of large-scale matrices,even for adjacency matrix A(G)which is(0,1)-symmetric matrices.Thus,many researchers have established a number of upper and lower bounds for some graphs to estimate this invariant.The aim of this paper is to study the relation between the energy and the edge domination number of G.The main contents are as follows.The first chapter introduces the background and current situation about the Graph energy.In chapter 2,we introduce some basic definitions and some known results related to our topic.In chapter 3,we consider the special case when the graph is dominated by just one edge.In chapter 4,we respectively establish the lower and the upper bound for graph energy by using the edge domination number of graphs.If G is a connected graph with edge domination number ?,then?(G)>2?,with equality if and only if G is the complete bipartite graph K?,?,and?(G)? 2?(?)+(?2-?)((?)+1)?,with equality if and only if G is the graph obtained from two copies of K1,?-1 by adding an edge joining their center vertices,where ? is the maximum vertex degree of G.