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A short note on $A_α$-eigenvalues for simple graphs

Giovanni Barbarino

TL;DR

This note investigates $A_\alpha$-eigenvalues of simple graphs by focusing on two lower bounds for the spectral radius $\lambda_1(A_\alpha)$: one depending on the maximum degree $\Delta$ and another on both $\Delta$ and the minimum degree $\delta$. It proves that when there are no isolated vertices ($\delta\ge 1$) the $\Delta$-$\delta$ bound is at least as strong as the $\Delta$-only bound, with equality in cases such as $\delta=1$ or $\alpha\in\{0,1\}$; for $\delta=0$ the inequality reverses. The authors provide a detailed algebraic comparison of the two bounds, introducing auxiliary variables to facilitate the inequality analysis, and they characterize precisely when one bound dominates the other or when they are equal. These results clarify which lower bound is preferable in different degree-structure regimes and contribute to sharper criteria for graph substructure (factors) via $A_\alpha$-spectral information.

Abstract

Given a simple graph $G$, its $A_α$ matrix is a convex combination with parameter $α\in [0,1]$ of its adjacency matrix and its degree diagonal matrices. Here we compare two lower bounds presented in [J. D. G. Silva Jr., C. S. Oliveira and L. M. G. C. Costa. "Some results involving the $A_α$-eigenvalues for graphs and line graphs"] for the spectral radius of $A_α$, and prove that one is better than the other when there are no isolated nodes in $G$.

A short note on $A_α$-eigenvalues for simple graphs

TL;DR

This note investigates -eigenvalues of simple graphs by focusing on two lower bounds for the spectral radius : one depending on the maximum degree and another on both and the minimum degree . It proves that when there are no isolated vertices () the - bound is at least as strong as the -only bound, with equality in cases such as or ; for the inequality reverses. The authors provide a detailed algebraic comparison of the two bounds, introducing auxiliary variables to facilitate the inequality analysis, and they characterize precisely when one bound dominates the other or when they are equal. These results clarify which lower bound is preferable in different degree-structure regimes and contribute to sharper criteria for graph substructure (factors) via -spectral information.

Abstract

Given a simple graph , its matrix is a convex combination with parameter of its adjacency matrix and its degree diagonal matrices. Here we compare two lower bounds presented in [J. D. G. Silva Jr., C. S. Oliveira and L. M. G. C. Costa. "Some results involving the -eigenvalues for graphs and line graphs"] for the spectral radius of , and prove that one is better than the other when there are no isolated nodes in .
Paper Structure (2 sections, 5 theorems, 20 equations)

This paper contains 2 sections, 5 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Main Results

Key Result

Proposition 1

2 If $G$ is a graph with maximum degree $\Delta$ and $\alpha\in [0,1]$, then If $G$ is connected, the equality holds if and only if $G\cong K_{1,\Delta}$.

Theorems & Definitions (9)

  • Proposition 1
  • Theorem 1
  • Conjecture 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Theorem 2
  • proof