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 $\lambda_{1} \geq \lambda_{2} \geq \ldots \geq \lambda_{n}$ 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: $$ E \geq \frac{2m}{\lambda_{1}}. $$ 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.}.