On the distribution of $A_α$-eigenvalues in terms of graph invariants
Uilton Cesar Peres Junior, Carla Silva Oliveira, André Ebling Brondan
TL;DR
The paper investigates the distribution of eigenvalues of the A_α(G) matrix, a convex combination $A_α(G) = α D(G) + (1-α) A(G)$ with α in [0,1]. It derives bounds on the number of A_α(G) eigenvalues in subintervals of the real line using graph invariants p(G), q(G), γ(G), β(G), and ν(G), and identifies extremal graph families that attain these bounds. The results extend known spectral bounds for the adjacency and signless Laplacian matrices and connect structural graph parameters to quantitative eigenvalue distribution, with explicit extremal examples such as stars, corona graphs, and certain path-derived families. Overall, the work broadens the understanding of how graph structure governs the spectrum of A_α(G) and provides tools for estimating eigenvalue distributions in chemical and network contexts.
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 $α\in [0,1]$, 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.