On the distribution of $A_\alpha$-eigenvalues in terms of graph invariants

Abstract

Let $G$ be a connected graph of order $n$, and $A(G)$ and $D(G)$ its adjacency and degree diagonal matrices, respectively. For a parameter $\alpha \in [0,1]$, Nikiforov~(2017) introduced the convex combination $A_{\alpha}(G) = \alpha D(G) + (1 - \alpha)A(G)$. In this paper, we investigate the spectral distribution of $A_\alpha(G)$-eigenvalues, over subintervals of the real line. We establish lower and upper bounds on the number of such eigenvalues in terms of structural parameters of $G$, including the number of pendant and quasi-pendant vertices, the domination number, the matching number, and the edge covering number. Additionally, we exhibit families of graphs for which these bounds are attained. Several of our results extend known spectral bounds on the eigenvalue distributions of both the adjacency and the signless Laplacian matrices.