New lower bounds for the energy of matrices and graphs

TL;DR

This paper introduces new lower bounds for the energy of Hermitian matrices and graphs, extending classical bounds and analyzing cases of equality, with implications for graphs with various nullities and edge counts.

Contribution

It generalizes existing lower bounds for graph energy to Hermitian matrices and graphs with specified nullity, providing new bounds and discussing equality cases.

Findings

New lower bounds for Hermitian matrix energy.

Bounds applicable to graphs with given nullity and edge count.

Some bounds are incomparable with the classical $2\sqrt{m}$ bound.

Abstract

Let $R$ be a Hermitian matrix. The energy of $R$, $\mathcal{E}(R)$, corresponds to the sum of the absolute values of its eigenvalues. In this work it is obtained two lower bounds for $\mathcal{E}(R).$ The first one generalizes a lower bound obtained by Mc Clellands for the energy of graphs in $1971$ to the case of Hermitian matrices and graphs with a given nullity. The second one generalizes a lower bound obtained by K. Das, S. A. Mojallal and I. Gutman in 2013 to symmetric non-negative matrices and graphs with a given nullity. The equality cases are discussed. These lower bounds are obtained for graphs with $m$ edges and some examples are provided showing that, some obtained bounds are incomparable with the known lower bound for the energy $2\sqrt{m}$. Another family of lower bounds are obtained from an increasing sequence of lower bounds for the spectral radius of a graph. The bounds are stated for singular and non-singular graphs.