A lower bound for the energy of graphs in terms of the vertex cover number

Abstract

The energy of the graph G, denoted by $E\left(G\right)$, is the sum of the absolute values of its eigenvalues. Wang and Ma proved that if G has c odd cycles, then $E\left(G\right)\ge 2\left(\beta \right(G)-c)$, where $\beta \left(G\right)$ is the vertex cover number of G. In this paper we strengthen this result by showing that if G and $\bar{G}$ have $c_o\left(G\right)$ and $c_o\left(\bar{G}\right)$ numbers of induced odd cycles, respectively, then $E\left(G\right)\ge 2\left(\beta \right(G)-\min \{c_o\left(G\right),c_o\left(\bar{G}\right)\left\}\right)$ and we conjecture that for every graph G, $E\left(G\right)\ge 2\beta \left(G\right)$. We prove the conjecture for some families of graphs, namely, bipartite graphs, $C_4$-free regular graphs, perfect graphs, and for all graphs with $\beta \left(G\right)\le \frac{\left|V\right(G\left)\right|}{2}$. It is shown that for every graph G, $2\left(\beta \right(G)-{\lambda }_1(\bar{G})-{\lambda }_n(\bar{G}\left)\right)\le E\left(G\right)$, where $\bar{G}$ is the complement of G, ${\lambda }_1\left(\bar{G}\right)$ and ${\lambda }_n\left(\bar{G}\right)$ denote the largest and the smallest eigenvalues of the adjacency matrix of $\bar{G}$, respectively. Using this we also prove that the conjecture holds for regular graphs with large degree.