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Some results involving the $A_α$-eigenvalues for graphs and line graphs

Joao Domingos Gomes da Silva Junior, Carla Silva Oliveira, Liliana Manuela Gaspar C. da Costa

TL;DR

This work studies spectral bounds for the $A_\alpha$-matrix, defined as $A_\alpha(G) = \alpha D(G) + (1-\alpha) A(G)$, and extends the analysis to the line graph via $A_\alpha(l(G))$. It derives two main lower bounds for the largest eigenvalue $\lambda_1(A_\alpha(G))$, plus a Rayleigh-type bound linked to degree-based indices, and provides an upper bound in terms of graph degrees and size with equality characterizations. The authors compare these bounds against Nikiforov's bounds across various graph families, identifying regimes where their bounds are tighter, as well as cases where bounds are incomparable. For line graphs, the paper establishes a lower bound on the smallest eigenvalue of $A_\alpha(l(G))$, exact results for regular graphs, and several bounds relating $\lambda_1(A_\alpha(l(G)))$ to the spectrum of $A_\alpha(G)$, thereby advancing the study of $A_\alpha$-spectra in both graphs and their line graphs.

Abstract

Let $G$ be a simple graph with adjacency matrix $A(G)$, signless Laplacian matrix $Q(G)$, degree diagonal matrix $D(G)$ and let $l(G)$ be the line graph of $G$. In 2017, Nikiforov defined the $A_α$-matrix of $G$, $A_α(G)$, as a linear convex combination of $A(G)$ and $D(G)$, the following way, $A_α(G):=αA(G)+(1-α)D(G),$ where $α\in[0,1]$. In this paper, we present some bounds for the eigenvalues of $A_α(G)$ and for the largest and smallest eigenvalues of $A_α(l(G))$. Extremal graphs attaining some of these bounds are characterized.

Some results involving the $A_α$-eigenvalues for graphs and line graphs

TL;DR

This work studies spectral bounds for the -matrix, defined as , and extends the analysis to the line graph via . It derives two main lower bounds for the largest eigenvalue , plus a Rayleigh-type bound linked to degree-based indices, and provides an upper bound in terms of graph degrees and size with equality characterizations. The authors compare these bounds against Nikiforov's bounds across various graph families, identifying regimes where their bounds are tighter, as well as cases where bounds are incomparable. For line graphs, the paper establishes a lower bound on the smallest eigenvalue of , exact results for regular graphs, and several bounds relating to the spectrum of , thereby advancing the study of -spectra in both graphs and their line graphs.

Abstract

Let be a simple graph with adjacency matrix , signless Laplacian matrix , degree diagonal matrix and let be the line graph of . In 2017, Nikiforov defined the -matrix of , , as a linear convex combination of and , the following way, where . In this paper, we present some bounds for the eigenvalues of and for the largest and smallest eigenvalues of . Extremal graphs attaining some of these bounds are characterized.
Paper Structure (9 sections, 46 theorems, 66 equations, 7 figures, 3 tables)

This paper contains 9 sections, 46 theorems, 66 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Some Bounds for text2-eigenvalues of Graphs
  5. Bounds Comparison
  6. Lower bounds proposed
  7. Lower bounds proposed and Nikiforov's bounds.
  8. Upper bounds
  9. Some results for text2-spectra of Line Graphs

Key Result

Theorem 2.1

horn2013matrix Let $A, B \in M_n({\rm I\!R})$ be symmetric and let the spectrum of $A$, $B$, and $A + B$ be $\sigma(A) = \{\lambda_1(A), \ldots, \lambda_n(A)\}$, $\sigma(B) = \{\lambda_1(B), \ldots, \lambda_n(B)\}$ and $\sigma(A+B) = \{\lambda_1(A+B), \ldots, \lambda_n(A+B)\}$, respectively. Then, for each $i = 1, \ldots, n$, with equality for some pair $i,j$ if and only if there is a nonzero vec

Figures (7)

  • Figure 1: Binomial tree.
  • Figure 2: A graph and its line graph.
  • Figure 3: Comparing lower bound for $W_n$.
  • Figure 4: Comparing lower bound for $K_{1,n-1}$.
  • Figure 5: Comparing lower bound for $W_n$.
  • ...and 2 more figures

Theorems & Definitions (70)

  • Theorem 2.1: Weyl
  • Corollary 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Proposition 2.10
  • ...and 60 more