On the eigenvalues and energy of the $A_{\alpha }$-matrix of graphs

Abstract

For a graph $G$, the generalized adjacency matrix $A_{\alpha }(G)$ is the convex combination of the diagonal matrix $D(G)$ and the adjacency matrix $A(G)$ and is defined as $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$ for $0\le \alpha \le 1$. This matrix has been found to be useful in merging the spectral theories of $A(G)$ and the signless Laplacian matrix $Q(G)$ of the graph $G$. The generalized adjacency energy or $A_{\alpha }$-energy is the mean deviation of the $A_{\alpha }$-eigenvalues of $G$ and is defined as $E(A_{\alpha }(G))={\sum }_{i=1}^n|p_i-\frac{2\alpha m}{n}|$, where $p_i$'s are $A_{\alpha }$-eigenvalues of $G$. In this paper, we investigate the $A_{\alpha }$-eigenvalues of a strongly regular graph $G$. We observe that $A_{\alpha }$-spectral radius $p_1$ satisfies $\delta (G)\le p_1\le \Delta (G)$, where $\delta (G)$ and $\Delta (G)$ are, respectively, the smallest and the largest degrees of $G$. Further, we show that the complete graph is the only graph to have exactly two distinct $A_{\alpha }$-eigenvalues. We obtain lower and upper bounds of $A_{\alpha }$-energy in terms of order, size and extremal degrees of $G$. We also discuss the extremal cases of these bounds.