Toughness and spectral radius in graphs

TL;DR

This paper establishes a spectral radius condition that characterizes when a connected graph is -tough, identifying a specific extremal graph structure when the spectral radius exceeds a certain threshold.

Contribution

It provides a spectral radius bound that guarantees -toughness in graphs, identifying the extremal case and a specific graph structure that nearly meets this bound.

Findings

Graphs with spectral radius above (t,n) are -tough unless they are a specific extremal graph.

The extremal graph for the spectral radius threshold is explicitly characterized.

A polynomial (t,n) determines the spectral radius threshold for -toughness.

Abstract

Let $t$ be a positive integer, and let $G$ be a connected graph of order $n$ with $n\geq t+2$. A graph $G$ is said to be $\frac{1}{t}$-tough if $|S|\geq\frac{1}{t}c(G-S)$ for every subset $S$ of $V(G)$ with $c(G-S)\geq2$, where $c(G-S)$ is the number of connected components in $G-S$. The adjacency matrix of $G$ is denoted by $A(G)$. Let $\lambda_1(G)\geq\lambda_2(G)\geq\dots\geq\lambda_n(G)$ be the eigenvalues of $A(G)$. In particular, the eigenvalue $\lambda_1(G)$ is called the spectral radius of $G$. In this paper, we prove that $G$ is a $\frac{1}{t}$-tough graph unless $G=K_1\vee(K_{n-t-1}\cup tK_1)$ if $\lambda_1(G)\geq\eta(t,n)$, where $\eta(t,n)$ is the largest root of $x^{3}-(n-t-2)x^{2}-(n-1)x+t(n-t-2)=0$.