Extremal spectral radius and $g$-good $r$-component connectivity
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| Format: | Preprint |
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2024
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| _version_ | 1866916466269356032 |
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| author | Ding, Wenxiu Li, Dan Wang, Yu |
| author_facet | Ding, Wenxiu Li, Dan Wang, Yu |
| contents | For $F\subseteq V(G)$, if $G-F$ is a disconnected graph with at least $r$ components and each vertex $v\in V(G)\backslash F$ has at least $g$ neighbors, then $F$ is called a $g$-good $r$-component cut of $G$. The $g$-good $r$-component connectivity of $G$, denoted by $cκ_{g,r}(G)$, is the minimum cardinality of $g$-good $r$-component cuts of $G$. Let $\mathcal{G}_n^{k,δ}$ be the set of graphs of order $n$ with minimum degree $δ$ and $g$-good $r$-component connectivity $cκ_{g,r}(G)=k$. In the paper, we determine the extremal graphs attaining the maximum spectral radii among all graphs in $\mathcal{G}_n^{k,δ}$. A subset $F\subseteq V(G)$ is called a $g$-good neighbor cut of $G$ if $G-F$ is disconnected and each vertex $v\in V(G)\backslash F$ has at least $g$ neighbors. The $g$-good neighbor connectivity $κ_g(G)$ of a graph $G$ is the minimum cardinality of $g$-good neighbor cuts of $G$. The condition of $g$-good neighbor connectivity is weaker than that of $g$-good $r$-component connectivity, and there is no requirement on the number of components. As a counterpart, we also study similar problem for $g$-good neighbor connectivity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_01854 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extremal spectral radius and $g$-good $r$-component connectivity Ding, Wenxiu Li, Dan Wang, Yu Combinatorics For $F\subseteq V(G)$, if $G-F$ is a disconnected graph with at least $r$ components and each vertex $v\in V(G)\backslash F$ has at least $g$ neighbors, then $F$ is called a $g$-good $r$-component cut of $G$. The $g$-good $r$-component connectivity of $G$, denoted by $cκ_{g,r}(G)$, is the minimum cardinality of $g$-good $r$-component cuts of $G$. Let $\mathcal{G}_n^{k,δ}$ be the set of graphs of order $n$ with minimum degree $δ$ and $g$-good $r$-component connectivity $cκ_{g,r}(G)=k$. In the paper, we determine the extremal graphs attaining the maximum spectral radii among all graphs in $\mathcal{G}_n^{k,δ}$. A subset $F\subseteq V(G)$ is called a $g$-good neighbor cut of $G$ if $G-F$ is disconnected and each vertex $v\in V(G)\backslash F$ has at least $g$ neighbors. The $g$-good neighbor connectivity $κ_g(G)$ of a graph $G$ is the minimum cardinality of $g$-good neighbor cuts of $G$. The condition of $g$-good neighbor connectivity is weaker than that of $g$-good $r$-component connectivity, and there is no requirement on the number of components. As a counterpart, we also study similar problem for $g$-good neighbor connectivity. |
| title | Extremal spectral radius and $g$-good $r$-component connectivity |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.01854 |