Scattering number and $\tau$-toughness in graphs involving $A_\alpha$-spectral radius

Abstract

The scattering number $s(G)$ of graph $G=(V,E)$ is defined as $s(G)$=max\big\{$c(G-S)-|S|$\big\}, where the maximum is taken over all proper subsets $S\subseteq V(G)$, and $c(G-S)$ denotes the number of components of $G-S$. In 1988, Enomoto introduced a variation of toughness $\tau(G)$ of a graph $G$, which is defined by $\tau(G)$=min\big\{$\frac{|S|}{c(G-S)-1}$, $S\subseteq V(G)$ and $c(G-S)>1$\big\}. 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 $\tau(G)$), the stronger the connectivity of a graph $G$. The $A_{\alpha}$-spectral radius of $G$ is denoted by $\rho_{\alpha}(G)$. Using typical $A_{\alpha}$-spectral techniques and structural analysis, we present a sufficient condition such that $s(G)\leq 1$. This result generalizes the result of Chen, Li and Xu [Graphs Comb. 41 (2025)]. Furthermore, we establish a sufficient condition with respect to the $A_{\alpha}$-spectral radius for a graph to be $\tau$-tough. When $\alpha=\frac{1}{2}$, our result reduces to that of Chen, Li and Xu [Graphs Comb. 41 (2025)].