A lower bound for the energy of hypoenergetic and non hypoenergetic graphs

Abstract

Let $G$ be a simple undirected graph with $n$ vertices and $m$ edges. The energy of $G,$ $\mathcal{E}(G)$ corresponds to the sum of its singular values. This work obtains lower bounds for $\mathcal{E}(G)$ where one of them generalizes a lower bound obtained by Mc Clelland in $1971$ to the case of graphs with given nullity. An extension to the bipartite case is given and, in this case, it is shown that the lower bound $2\sqrt{m}$ is improved. The equality cases are characterized. Moreover, a simple lower bound that considers the number of edges and the diameter of $G$ is derived. A simple lower bound, which improves the lower bound $2\sqrt{n-1}$, for the energy of trees with $n$ vertices and diameter $d$ is also obtained.