Graph energy estimates via the Chebyshev functional

TL;DR

This paper introduces new lower bounds for the energy of a graph based on Chebyshev functional analysis, improving existing bounds especially for certain classes of graphs like triangle-free graphs.

Contribution

It derives novel lower bounds on graph energy using Chebyshev functional techniques, enhancing previous results and providing simpler, more effective inequalities.

Findings

New lower bounds on graph energy are established.

The bounds improve upon known results for specific graph classes.

A simple inequality E ≥ 2m/λ₁ is derived, strengthening prior bounds.

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.}.