2025/09/01 by Caili Jia, Yong Lu, Jia, Caili +1
Mathematics · Computer Science · Physics and Astronomy · #Graph theory and applications #Graph Labeling and Dimension Problems #Complex Network Analysis Techniques
paper · pdf · doi:10.48550/arxiv.2509.01050
The scattering number s(G) of graph G=(V,E) is defined as s(G)=max\c(G-S)-|S|\, where the maximum is taken over all proper subsets S⊆ V(G), and c(G-S) denotes the number of components of G-S. In 1988, Enomoto introduced a variation of toughness τ(G) of a graph G, which is defined by τ(G)=min\(|S|)/(c(G-S)-1), S⊆ V(G) and c(G-S)>1\. Both the scattering number and toughness are used to characterize the invulnerability or stability of a graph, i.e., the ability of a graph to remain connected after vertices or edges are removed. The smaller the value of s(G) (or the larger the value of τ(G)), the stronger the connectivity of a graph G. The Aα-spectral radius of G is denoted by ρα(G). Using typical Aα-spectral techniques and structural analysis, we present a sufficient condition such that s(G)≤ 1. This result generalizes the result of Chen, Li and Xu [Graphs Comb. 41 (2025)]. Furthermore, we establish a sufficient condition with respect to the Aα-spectral radius for a graph to be τ-tough. When α=(1)/(2), our result reduces to that of Chen, Li and Xu [Graphs Comb. 41 (2025)].