ON SOME BOUNDS OF THE MINIMUM EDGE DOMINATING ENERGY OF A GRAPH

Abstract

Let G be a simple graph of order n with vertex set V= {v₁, v₂, ..., v_n} and edge set  E = {e₁, e₂, ..., e_m}. A subset  of E is called an edge dominating set of G if every edge of  E -  is adjacent to some edge in  .Any edge dominating set with minimum cardinality is called a minimum edge dominating set [2]. Let  be a minimum edge dominating set of a graph G. The minimum edge dominating matrix of G is the m x m matrix defined by

G)= , where  =

The characteristic polynomial of is denoted by

f_n (G, ρ) = det (ρI -  (G) ). 

The minimum edge dominating eigen values of a graph G are the eigen values of (G).  Minimum edge dominating energy of G is defined as

                (G) =   [12]

In this paper we have computed the Minimum Edge Dominating Energy of a graph. Its properties and bounds are discussed. All graphs considered here are simple, finite and undirected.