Some new bounds for the energy of graphs

Abstract

Let $G$ be a graph with $n$ vertices and $m$ edges. The energy of a graph $G$ is defined as the sum of absolute values of the eigenvalues about its adjacency matrix, i.e. $\varepsilon \left(G\right)={\sum }_{i=1}^n\left|{\lambda }_i\right|$. In this paper, we derive some new upper bounds on the graph energy based on a new formula and some inequalities for calculating the graph energy, and characterize the extremal graphs. In addition, we propose some new lower bounds for the graph energy involving order $n$, the size $m$, the eigenvalue with maximum absolute value ${\lambda }_1$ and the eigenvalue with minimum absolute value ${\lambda }_n$ of the graph $G$, and characterize the extremal graphs.