Sufficient conditions for the variation of toughness under the distance spectral in graphs involving minimum degree

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.