Scattering number and $τ$-toughness in graphs involving $A_α$-spectral radius
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908512159793152 |
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| author | Jia, Caili Lu, Yong |
| author_facet | Jia, Caili Lu, Yong |
| contents | 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 $τ(G)$ of a graph $G$, which is defined by $τ(G)$=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 $τ(G)$), the stronger the connectivity of a graph $G$. The $A_α$-spectral radius of $G$ is denoted by $ρ_α(G)$. Using typical $A_α$-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 $A_α$-spectral radius for a graph to be $τ$-tough. When $α=\frac{1}{2}$, our result reduces to that of Chen, Li and Xu [Graphs Comb. 41 (2025)]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01050 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scattering number and $τ$-toughness in graphs involving $A_α$-spectral radius Jia, Caili Lu, Yong Combinatorics 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 $τ(G)$ of a graph $G$, which is defined by $τ(G)$=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 $τ(G)$), the stronger the connectivity of a graph $G$. The $A_α$-spectral radius of $G$ is denoted by $ρ_α(G)$. Using typical $A_α$-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 $A_α$-spectral radius for a graph to be $τ$-tough. When $α=\frac{1}{2}$, our result reduces to that of Chen, Li and Xu [Graphs Comb. 41 (2025)]. |
| title | Scattering number and $τ$-toughness in graphs involving $A_α$-spectral radius |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.01050 |