On $\alpha$-adjacency energy of graphs and Zagreb index

Abstract

Let $A(G)$ be the adjacency matrix and $D(G)$ be the diagonal matrix of the vertex degrees of a simple connected graph $G$. Nikiforov defined the matrix $A_{\alpha }(G)$ of the convex combinations of $D(G)$ and $A(G)$ as $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$, for $0\le \alpha \le 1$. If ${\rho }_1\ge {\rho }_2\ge \cdots \ge {\rho }_n$ are the eigenvalues of $A_{\alpha }(G)$ (which we call $\alpha$-adjacency eigenvalues of $G$), the $\alpha$-adjacency energy of $G$ is defined as $E^{A_{\alpha }}(G)={\sum }_{i=1}^n\left|{\rho }_i-\frac{2\alpha m}{n}\right|$, where $n$ is the order and $m$ is the size of $G$. We obtain the upper and lower bounds for $E^{A_{\alpha }}(G)$ in terms of order $n$, size $m$ and Zagreb index $Zg(G)$ associated to the structure of $G$. Further, we characterize the extremal graphs attaining these bounds.

17 pages