Graph energy estimates via the Chebyshev functional

Abstract

Let $G$ be a graph with $n$ vertices and $m$ edges. The energy $E$ of the graph $G$ is defined as the sum of the moduli of the adjacency eigenvalues ${\lambda }_1\ge {\lambda }_2\ge \dots \ge {\lambda }_n$ of $G$: $$E=\sum _{i=1}^n|\lambda i|.$$ We obtain new lower bounds on the energy of a graph, which in various cases improve upon known results. For example, a particularly simple and appealing corollary of our results is: $$E\ge \frac{2m}{{\lambda }_1}.$$ This implies a result obtained by Gutman \emph{et al.} for regular graphs and is better for triangle-free graphs than a result of Caporossi \emph{et al.}.