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The existence of even factors based on the $A_α$-spectral radius of graphs

Caili Jia, Yong Lu

TL;DR

"The existence of even factors based on the $A_{α}$-spectral radius of graphs" studies when a connected graph with minimum degree $δ$ and even order contains an even factor by a spectral condition on the $A_{α}$-matrix. It generalizes prior results by using the $A_{α}$-spectral radius and identifies the extremal join graph $K_{δ} \vee (K_{n-2δ+1} \cup (δ-1)K_{1})$ as the benchmark; if ρ_{α}(G) is at least the bound, G has an even factor unless G is the extremal graph. The proof relies on a sequence of lemmas about $ρ_{α}$, equitable partitions, and Perron vectors, together with a careful case analysis and graph-transform arguments to rule out counterexamples. The results extend known α-parameterized conditions and deepen the link between spectral radius and spanning subgraph structure.

Abstract

An even factor of $G$ is a spanning subgraph $F$ such that every vertex in $F$ has a nonzero even degree. Note that $δ(G)\geq2$ is a trivial necessary condition for a graph to have an even factor, where $δ(G)$ is the minimum degree of $G$. In this paper, for a connected graph $G$ with minimum degree $δ$, we establish a lower bound on the $A_α$-spectral radius of $G$ such that $G$ contains an even factor.

The existence of even factors based on the $A_α$-spectral radius of graphs

TL;DR

"The existence of even factors based on the -spectral radius of graphs" studies when a connected graph with minimum degree and even order contains an even factor by a spectral condition on the -matrix. It generalizes prior results by using the -spectral radius and identifies the extremal join graph as the benchmark; if ρ_{α}(G) is at least the bound, G has an even factor unless G is the extremal graph. The proof relies on a sequence of lemmas about , equitable partitions, and Perron vectors, together with a careful case analysis and graph-transform arguments to rule out counterexamples. The results extend known α-parameterized conditions and deepen the link between spectral radius and spanning subgraph structure.

Abstract

An even factor of is a spanning subgraph such that every vertex in has a nonzero even degree. Note that is a trivial necessary condition for a graph to have an even factor, where is the minimum degree of . In this paper, for a connected graph with minimum degree , we establish a lower bound on the -spectral radius of such that contains an even factor.

Paper Structure

This paper contains 3 sections, 9 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminary lemmas
  3. Proof of Theorem \ref{['th:1.1.']}

Key Result

Theorem 1.1

Let $\alpha\in[0,1)$, and let $G$ be a connected graph with minimum degree $\delta\geq 2$ and even order If $\rho_{\alpha}(G)\geq\rho_{\alpha}(K_{\delta}\vee(K_{n-2\delta+1}\cup{(\delta-1)} K_{1})),$ then $G$ contians an even factor, unless $G\cong K_{\delta}\vee(K_{n-2\delta+1}\cup{(\delta-1)} K_{1})$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8