Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounds on $A_α$-eigenvalues using graph invariants

João Domingos Gomes da Silva Junior, Carla Silva Oliveira, Liliana Manuela Gaspar Cerveira da Costa

TL;DR

The paper addresses bounds on the extrema of the $A_\alpha$-spectrum by relating $A_\alpha$ to graph invariants. It develops multiple analytic bounds via Rayleigh quotients, equitable partitions, Zagreb indices, and comparisons to $A(G)$ and $Q(G)$, covering both the largest and smallest eigenvalues. Key contributions include new upper bounds for $\lambda_1(A_\alpha(G))$ expressed through $n,m,\Delta,\Delta_2, diam(G), Z_1(G), \omega$, and a lower/upper bound for $\lambda_n(A_\alpha(G))$ with equality characterizations in regular or semiregular graphs. These results deepen understanding of how graph structure governs the $A_\alpha$-spectrum and may inform spectral graph theory applications.

Abstract

In 2017, Nikiforov introduced the concept of the $A_α$-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 $A_α$-matrix involving invariants associated to graphs.

Bounds on $A_α$-eigenvalues using graph invariants

TL;DR

The paper addresses bounds on the extrema of the -spectrum by relating to graph invariants. It develops multiple analytic bounds via Rayleigh quotients, equitable partitions, Zagreb indices, and comparisons to and , covering both the largest and smallest eigenvalues. Key contributions include new upper bounds for expressed through , and a lower/upper bound for with equality characterizations in regular or semiregular graphs. These results deepen understanding of how graph structure governs the -spectrum and may inform spectral graph theory applications.

Abstract

In 2017, Nikiforov introduced the concept of the -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 -matrix involving invariants associated to graphs.
Paper Structure (3 sections, 22 theorems, 64 equations, 1 figure)

This paper contains 3 sections, 22 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results

Key Result

Theorem 2.1

ADB_Equitable Let $M$ be a square matrix of order $n$ and suppose that $M$ has an equitable partition $\pi = \{P_1,P_2,\ldots,P_k\}$. Let $N$ be the quotient matrix of $M$ with respect to the partition $\pi$. Then the eigenvalues of $N$ are eigenvalues of $M$.

Figures (1)

  • Figure 1: Examples of graphs to illustrate the concepts of $\Delta$, $d_2$ and $\Delta_2$

Theorems & Definitions (30)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Lemma 2.10
  • ...and 20 more