2025/04/10 by Liu, Xiangge, Lu, Yong, Jia, Caili +2
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.07501
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)≤|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≥ 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 (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.