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

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 $0\leq \alpha \leq 1$. 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 $\delta(G)\leq p_1 \leq \Delta(G)$, 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.