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arXiv · 1903.01326

New lower bounds for the energy of matrices and graphs

Enide Andrade·Juan Carmona·Geraldine Infante·María Robbiano
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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.

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Enide Andrade, Juan Carmona, Geraldine Infante, María Robbiano. 2019-03-04. New lower bounds for the energy of matrices and graphs. https://arxiv.org/abs/1903.01326

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