Graph energy estimates via the Chebyshev functional

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 $λ_{1} \geq λ_{2} \geq \ldots \geq λ_{n}$ of $G$: $E=\sum_{i=1}^{n}{|λ{i}|}.$ 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: $E \geq \frac{2m}{λ_{1}}.$ 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.}.

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