Distance signless Laplacian spectral radius and tough graphs involving minimun degree

TL;DR

The paper develops spectral conditions based on the distance signless Laplacian radius $\\eta_{1}(G)$ to guarantee $t$-toughness of graphs with lower bounds on minimum degree. By comparing $\\eta_{1}(G)$ to those of extremal join-graphs and employing equitable partitions, the authors establish sufficiency results for $1$-toughness with minimum degree and for $t$-toughness when $1/t$ is a positive integer, plus bounds expressed in terms of the order $n$ and the edge count $m$. The results strengthen the link between distance-based spectral parameters, graph toughness, and structural extremality, and they provide explicit exceptional graphs for equality cases. Collectively, these findings offer practical spectral criteria for assessing toughness in graphs with prescribed minimum degree and size, with implications for Hamiltonicity-related properties.

Abstract

Let $G=(V(G),E(G))$ be a simple graph, where $V(G)$ and $E(G)$ are the vertex set and the edge set of $G$, respectively. The number of components of $G$ is denoted by $c(G)$. Let $t$ be a positive real number, and a connected graph $G$ is $t$-tough if $t c(G-S)\leq|S|$ for every vertex cut $S$ of $V(G)$. The toughness of graph $G$, denoted by $τ(G)$, is the largest value of $t$ for which $G$ is $t$-tough. Recently, Fan, Lin and Lu [European J. Combin. 110(2023), 103701] presented sufficient conditions based on the spectral radius for graphs to be 1-tough with minimum degree $δ(G)$ and graphs to be $t$-tough with $t\geq 1$ being an integer, respectively. In this paper, we establish sufficient conditions in terms of the distance signless Laplacian spectral radius for graphs to be 1-tough with minimum degree $δ(G)$ and graphs to be $t$-tough, where $\frac{1}{t}$ is a positive integer. Moreover, we consider the relationship between the distance signless Laplacian spectral radius and $t$-tough graphs in terms of the order $n$.

Figures (2)

• Figure 1: $K_{\delta}\vee(K_{n-2\delta}+\delta K_{1})$ .
• Figure 2: $K_{1}\vee(K_{n-\frac{1}{t}-1}+\frac{1}{t}K_{1})$.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 3.1
• Lemma 4.1