🤖 AI Summary
This study characterizes the distribution of eigenvalues of the $A_alpha$-matrix of a graph $G$ over real subintervals, aiming to establish quantitative bounds—both upper and lower—on the number of such eigenvalues in terms of structural graph parameters, including the number of pendant vertices, quasi-pendant vertices, domination number, matching number, and edge covering number. Employing algebraic graph theory techniques—including analysis of the characteristic polynomial, Rayleigh quotient estimation, and structural induction—we systematically relate the $A_alpha$-spectrum distribution to multiple graph invariants for the first time. This unifies and generalizes classical spectral bounds for both the adjacency matrix and the signless Laplacian matrix. The derived bounds are tight, and extremal graph families achieving these bounds are explicitly constructed, thereby confirming the optimality and broad applicability of the theoretical results.
📝 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 $αin [0,1]$, Nikiforov~(2017) introduced the convex combination $A_α(G) = αD(G) + (1 - α)A(G)$. In this paper, we investigate the spectral distribution of $A_α(G)$-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.