Some properties of maximum deficiency energy of a graph

摘要

The concept of maximum deficiency matrix M-df (G) of a simple graph G is introduced in this paper. Let G = (V, E) be a simple graph of order n and let df(v(i)) be the deficiency of a vertex v(i), i = 1, 2, . . . , n, then the maximum deficiency matrix M-df (G) = [f(ij)](nxn) is defined as:f(ij) = {max{df(v(i)), df(v(j))}, if v(i)v(j) is an element of E(G) 0, otherwise.Further, some coefficients of the characteristic polynomial phi(G; gamma) of the maximum deficiency matrix of G are obtained. The maximum deficiency energy EMdf (G) of a graph G is also introduced. The bounds for EMdf (G) are established. Moreover, maximum deficiency energy of some standard graphs is shown, and if the maximum deficiency energy of a graph is rational, then it must be an even integer.