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Signless Laplacian spectral conditions: Forbidden $4$-cycle and star embeddings

Zhe Wei, Zhenzhen Lou, Changxiang He

Abstract

The signless Laplacian spectral radius has emerged as a crucial spectral parameter in network science. This paper establishes new extremal results in spectral graph theory by investigating the signless Laplacian spectral radius ($Q$-index) of graphs with forbidden subgraphs. We present a $Q$-spectral analog of classical Nosal-type theorems, providing sharp conditions that guarantee the existence of either a $4$-cycle or a large star $K_{1,m-k}$ in a graph. The main theorem states that for integers $k \geq 0$ and graphs $G$ with size $m \geq \max\{7k+31, k^2+8(k+1)\}$, if $q(G) \geq q(S^+_{m,k+1})$, then $G$ must contain a $4$-cycle or $K_{1,m-k}$, unless $G$ is isomorphic to the extremal graph $S^+_{m,k+1}$ formed by adding $k+1$ independent edges to the star $K_{1,m-k-1}$. This result refines previous work on star embeddings by Wang and Guo [Journal of Algebraic Combinatorics, 59 (2024) 213--224], and completes the $Q$-spectral counterpart to Wang's adjacency spectral theorem for $4$-cycle containment [Discrete Math., 345 (2022) 112973]. Our analysis reveals new insights into how signless Laplacian eigenvalues encode graph structure, with tight bounds demonstrated through explicit extremal graph constructions and asymptotic analysis.

Signless Laplacian spectral conditions: Forbidden $4$-cycle and star embeddings

Abstract

The signless Laplacian spectral radius has emerged as a crucial spectral parameter in network science. This paper establishes new extremal results in spectral graph theory by investigating the signless Laplacian spectral radius (-index) of graphs with forbidden subgraphs. We present a -spectral analog of classical Nosal-type theorems, providing sharp conditions that guarantee the existence of either a -cycle or a large star in a graph. The main theorem states that for integers and graphs with size , if , then must contain a -cycle or , unless is isomorphic to the extremal graph formed by adding independent edges to the star . This result refines previous work on star embeddings by Wang and Guo [Journal of Algebraic Combinatorics, 59 (2024) 213--224], and completes the -spectral counterpart to Wang's adjacency spectral theorem for -cycle containment [Discrete Math., 345 (2022) 112973]. Our analysis reveals new insights into how signless Laplacian eigenvalues encode graph structure, with tight bounds demonstrated through explicit extremal graph constructions and asymptotic analysis.
Paper Structure (3 sections, 9 theorems, 47 equations)

This paper contains 3 sections, 9 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Proof of Theorem \ref{['thm-2']}

Key Result

Theorem 1.1

Let $G$ be a graph of size $m \geq 27$. If $\rho(G) \geq \sqrt{m-1}$, then $G$ contains a $C_4$, unless $G$ is one of the following graphs (possibly with isolated vertices): $K_{1,m}$, $K_{1,m-1+e}$, $K^e_{1,m-1}$, or $K_{1,m-1} \cup P_2$.

Theorems & Definitions (31)

  • Theorem 1.1: Wang-2
  • Theorem 1.2
  • Theorem 1.3: Zhai-1
  • Theorem 1.4: Wang-1
  • Theorem 1.5
  • Lemma 2.1: Cvetkovic1
  • Lemma 2.2
  • proof
  • Lemma 2.3: Feng
  • Lemma 2.4
  • ...and 21 more