Remarks on the energy of regular graphs

Abstract

The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. This note is about the energy of regular graphs. It is shown that graphs that are close to regular can be made regular with a negligible change of the energy. Also a k-regular graph can be extended to a k-regular graph of a slightly larger order with almost the same energy. As an application, it is shown that for every sufficiently large n, there exists a regular graph G of order n whose energy G * satisfies G * > 1 2 n 3/2 − n 13/10 .Several infinite families of graphs with maximal or submaximal energy are given, and the energy of almost all regular graphs is determined.

Barabasi-Albert trees are hypoenergetic

“…To the best of our knowledge the only papers considering the energy of a random graph are due Nikiforov [14,15]. In these papers, Nikiforov describes precisely the asymptotic behavior, as the size of the graph goes to infinity, of the energy of two families of random graphs: Erdös-Rényi graph with fixed p, [14], and uniform d-regular graphs, [15]. Both of these results rely on the fact that the explicit limiting distributions of the adjacency matrix of these graphs are well known.…”

Energy of a vertex

“…In other direction, as pointed out by Nikiforov [21,22], the energy of random graphs either for an Erdos-Renyi graph or for uniform d-regular graphs may be deduced easily from well-known results from Random Matrix Theory. This results may be translated directly to a typical vertex in a random graph.…”

A new method for the sum-edge characteristic polynomials of graphs

“…Das studied maximum energy of bipartite graphs in [10]. In [13], energy of regular graphs is considered. Naturally, being a linear algebraic concept, graph energy is related to many other notions related to graphs.…”