A lower bound of toughness of regular graphs: in terms of second largest eigenvalue

TL;DR

This paper establishes a lower bound on the toughness of connected regular graphs based on their second largest eigenvalue, linking spectral properties to graph connectivity resilience.

Contribution

It provides a new spectral bound on the toughness of regular graphs, connecting eigenvalues with graph robustness measures.

Findings

Toughness is at least rac{d+1}{d}(d- ext{second largest eigenvalue}) or 1, whichever is smaller.

The bound applies to connected, non-complete d-regular graphs with d ≥ 3.

Spectral properties can be used to estimate graph toughness and resilience.

Abstract

Let $G$ be a connected (non-complete) $d$-regular graph with $d\geq3$. Let $c(G-S)$ denote the number of components of $G-S$ for any cut $S$ of $G$. The toughness $t(G)$ of $G$ is defined as $\min\left\{\frac{|S|}{c(G-S)}\right\}$, where the minimum is taken over all proper cuts $S$ of $G$. Let $\lambda_{2}(G)$ denote the second largest eigenvalue of $G$. In this paper, we prove $$t(G)\geq\min\left\{\frac{d+1}{d}(d-\lambda_{2}(G)),1\right\}.$$