Generalized toughness and Q-index in a graph

2025/10/12 by Zhou, Sizhong
#05C35 #05C50 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.10498

Abstract

Let G be a graph. We denote by c(G), α(G) and q(G) the number of components, the independence number and the signless Laplacian spectral radius (Q-index for short) of G, respectively. The toughness of G is defined by t(G)=min\(|S|)/(c(G-S)):S⊆ V(G), c(G-S)≥2\ for G≠ K_n and t(G)=+∞ for G=K_n. Chen, Gu and Lin [Generalized toughness and spectral radius of graphs, Discrete Math. 349 (2026) 114776] generalized this notion and defined the l-toughness t_l(G) of a graph G as t_l(G)=min\(|S|)/(c(G-S)):S⊂ V(G), c(G-S)≥ l\ if 2≤ l≤α(G), and t_l(G)=+∞ if l>α(G). If t_l(G)≥ t, then G is said to be (t,l)-tough. In this paper, we put forward Q-index conditions for a graph to be (b,l)-tough and ((1)/(b),l)-tough, respectively.

Citations

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