The energy of random graphs

TL;DR

This paper studies the energy of random graphs, providing exact estimates using Wigner's semi-circle law and extending results to random multipartite graphs, advancing understanding of spectral graph invariants.

Contribution

It introduces a novel application of Wigner's semi-circle law to estimate the energy of almost all graphs and generalizes this to random multipartite graphs.

Findings

Exact energy estimates for almost all graphs using Wigner's law

Extension of energy bounds to random multipartite graphs

Improved understanding of spectral properties of random graphs

Abstract

In 1970s, Gutman introduced the concept of the energy $\En(G)$ for a simple graph $G$, which is defined as the sum of the absolute values of the eigenvalues of $G$. This graph invariant has attracted much attention, and many lower and upper bounds have been established for some classes of graphs among which bipartite graphs are of particular interest. But there are only a few graphs attaining the equalities of those bounds. We however obtain an exact estimate of the energy for almost all graphs by Wigner's semi-circle law, which generalizes a result of Nikiforov. We further investigate the energy of random multipartite graphs by considering a generalization of Wigner matrix, and obtain some estimates of the energy for random multipartite graphs.