Sufficient conditions for the variation of toughness under the distance spectral in graphs involving minimum degree
Peishan Li
TL;DR
This work connects the variation of toughness τ(G) to the distance spectral radius by establishing sufficient conditions that involve the minimum degree δ and the distance spectral radius λ1(D(G)). Building on prior results linking distance spectra to toughness, it derives extremal join-graph constructions G* and uses equitable partitions, Wiener-index bounds, and polynomial spectral comparisons to show that, under λ1(D(G)) ≤ λ1(D(G*)), a connected graph must be τ-tough unless it is isomorphic to G*. The results provide spectral criteria for τ-toughness and tie graph structure to distance-based spectral properties, with implications for factor existence and graph robustness.
Abstract
The concept of graph toughness was first introduced in 1973. In 1995, scholars first explored the lower bound of the toughness of connected d-regular graphs with respect to d and the second largest eigenvalue of the adjacency matrix. The concept of the variation of toughness was first introduced in 1988. The variation of toughness is defined as tau(G) = min{|S|/(c(G-S)-1)}. In 2025, Chen, Fan, and Lin provided sufficient conditions for a graph to be t-tough in terms of the minimum degree and the distance spectral radius. Inspired by this, we propose a sufficient condition for a graph to be tau-tough in terms of minimum degree and distance spectral radius, and provide the corresponding proof, where |S| and c(G-S)-1 are mutually divisible.