Improved bound of graph energy in terms of vertex cover number

Let $G$ be a simple graph with the vertex cover number $\tau$. The energy $\mathcal{E}(G)$ of $G$ is the sum of the absolute values of all the adjacency eigenvalues of $G$. In this article, we establish $\mathcal{E}(G)\ge 2\tau$ for several classes of graphs. The result significantly improves the known result $\mathcal{E}(G)\ge 2\tau -2c$ for many classes of graphs, where $c$ is the number of odd cycles.