Some bounds on the $A_\alpha$-energy of graphs

Abstract

Let $G$ be a graph with order $n$ and size $m$. For any real number $\alpha\in [0,1]$, Nikiforov defined the matrix $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the diagonal matrix of the vertex degrees. Let $\rho_i~(i=1, 2, \ldots, n)$ denote the eigenvalues of $A_\alpha(G)$ and $E^{A_\alpha}(G)=\sum\nolimits_{i=1}^n|\rho_i-\frac{2\alpha m}{n}|$ denote the $A_\alpha$-energy of $G$. In this paper, we get some lower bounds of $E^{A_\alpha}(G)$ in terms of the order, the size and the first Zagreb index of $G$ for $\alpha\in[\frac{1}{2},1)$, and characterize the extremal graphs when attaining the bounds if $G$ is regular. In addition, we give some lower and upper bounds of  $E^{A_\alpha}(G)$ under the condition that $\rho_1+\rho_n\geq\frac{4\alpha m}{n}$.