Bounds on $A_\alpha$-eigenvalues using graph invariants

TL;DR

This paper establishes bounds on the extremal eigenvalues of the $A_\alpha$-matrix, a convex combination of adjacency and degree matrices, using various graph invariants.

Contribution

It provides new bounds for the largest and smallest $A_\alpha$-eigenvalues based on graph invariants, enhancing understanding of spectral properties.

Findings

Derived bounds for the largest $A_\alpha$-eigenvalue.

Derived bounds for the smallest $A_\alpha$-eigenvalue.

Unified approach connecting eigenvalues with graph invariants.

Abstract

In 2017, Nikiforov introduced the concept of the $A_{\alpha}$-matrix, as a linear convex combination of the adjacency matrix and the degree diagonal matrix of a graph. This matrix has attracted increasing attention in recent years, as it serves as a unifying structure that combines the adjacency matrix and the signless Laplacian matrix. In this paper, we present some bounds for the largest and smallest eigenvalue of $A_{\alpha}$-matrix involving invariants associated to graphs.