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The maximum spectral radius of $θ_{1,3,3}$-free graphs with given size

Jing Gao, Xueliang Li

Abstract

A graph $G$ is said to be $F$-free if it does not contain $F$ as a subgraph. A theta graph, say $θ_{l_1,l_2,l_3}$, is the graph obtained by connecting two distinct vertices with three internally disjoint paths of length $l_1, l_2, l_3$, where $l_1\leq l_2\leq l_3$ and $l_2\geq2$. Recently, Li, Zhao and Zou [arXiv:2409.15918v1] characterized the $θ_{1,p,q}$-free graph of size $m$ having the largest spectral radius, where $q\geq p\geq3$ and $p+q\geq2k+1\geq7$, and proposed a problem on characterizing the graphs with the maximum spectral radius among $θ_{1,3,3}$-free graphs. In this paper, we consider this problem and determine the maximum spectral radius of $θ_{1,3,3}$-free graphs with size $m$ and characterize the extremal graph. Up to now, all the graphs in $\mathcal{G}(m,θ_{1,p,q})$ which have the largest spectral radius have been determined, where $q\geq p\geq 2$.

The maximum spectral radius of $θ_{1,3,3}$-free graphs with given size

Abstract

A graph is said to be -free if it does not contain as a subgraph. A theta graph, say , is the graph obtained by connecting two distinct vertices with three internally disjoint paths of length , where and . Recently, Li, Zhao and Zou [arXiv:2409.15918v1] characterized the -free graph of size having the largest spectral radius, where and , and proposed a problem on characterizing the graphs with the maximum spectral radius among -free graphs. In this paper, we consider this problem and determine the maximum spectral radius of -free graphs with size and characterize the extremal graph. Up to now, all the graphs in which have the largest spectral radius have been determined, where .

Paper Structure

This paper contains 3 sections, 7 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['main theorem']}

Key Result

Theorem 1.1

Li-Zhai-2024 Let $k\geq3$ and $m\geq4(k^2+3k+1)^2$. If $G\in\mathcal{G}(m,\theta_{1,2,2k-1})\cup \mathcal{G}(m,\theta_{1,2,2k})$, then and equality holds if and only if $G\cong K_k\vee(\frac{m}{k}-\frac{k-1}{2})K_1$.

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Claim 3.2
  • Claim 3.3
  • Claim 3.4