Status sum eigenvalues and energy of graphs

Abstract

Let G be a simple connected graph with vertex set V(G). The status of a vertex v ∈ V(G) is denoted by σ(v) and defined as the sura of distances from t; to all other vertices of G. The status sum matrix of G is defined by Sσ(G) = [sij] where Sij= σ(vi) + σ(vj) if i≠ j, and sij= 0 otherwise. The status sum energy Eσ(G) of a graph G is the sum of the absolute values of the eigenvalues of the status sum matrix. In this paper, we obtain some coefficients of the characteristic polynomial Φ(G, μ) of status sum matrix. Status sum energies of some well known graphs are obtained and some upper and lower bounds for Eσ(G) are established.