Bounds for a alpha-eigenvalues

TL;DR

This paper derives new bounds for the eigenvalues of the Aalpha-matrix, a convex combination of a graph's adjacency and degree matrices, and characterizes extremal graphs that attain these bounds.

Contribution

It introduces novel upper and lower bounds for the eigenvalues of the Aalpha-matrix and characterizes extremal graphs achieving these bounds.

Findings

New bounds for the largest, second largest, and smallest eigenvalues of Aalpha(G)

Characterization of extremal graphs attaining the bounds

Enhanced understanding of spectral properties of Aalpha-matrices

Abstract

Let G be a graph with adjacency matrix A(G) and degree diagonal matrix D(G). In 2017, Nikiforov [1] defined the matrix Aalpha(G), as a convex combination of A(G) and D(G), the following way, Aalpha(G) = alpha A(G) + (1 - alpha)D(G), where alpha belongs to [0,1]. In this paper, we present some new upper and lower bounds for the largest, second largest, and smallest eigenvalue of the Aalpha-matrix. Moreover, extremal graphs attaining some of these bounds are characterized