On graph energy, maximum degree and vertex cover number

Abstract

For a simple graph $G$ with $n$ vertices and $m$ edges having adjacency eigenvalues $\lambda_1,\lambda_2, \dots,\lambda_n$, the energy $E(G)$ of $G$ is defined as $E(G)=\sum_{i=1}^{n} |\lambda_i|$. We obtain the upper bounds for $E(G)$ in terms of the vertex covering number $\tau$, the number of edges $m$, maximum vertex degree $d_1$ and second maximum vertex degree $d_2$ of the connected graph $G$. These upper bounds improve some recently known upper bounds for $E(G)$. Further, these upper bounds for $E(G)$ imply a natural extension to other energies like distance energy and Randi\'{c} energy associated to a connected graph $G$.