New Upper Bounds for the Energy and Spectral Radius of Graphs

Let G be a finite simple undirected graph with n vertices and m edges. The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. In this paper we give some new upper bounds for E(G) in terms of n, m, the largest and the smallest eigenvalue, and the standard deviation of the squared eigenvalues of G. Moreover, we present an upper bound for the spectral radius of G in terms of n,m and E(G). New upper bound for the energy of the reciprocal graphs is also obtained. A number of our results rely on the use of well-known inequalities which have not been applied in this area before.