Extremal spectral radius and $g$-good $r$-component connectivity

Abstract

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 $c\kappa_{g,r}(G)$, is the minimum cardinality of $g$-good $r$-component cuts of $G$. Let $\mathcal{G}_n^{k,\delta}$ be the set of graphs of order $n$ with minimum degree $\delta$ and $g$-good $r$-component connectivity $c\kappa_{g,r}(G)=k$. In the paper, we determine the extremal graphs attaining the maximum spectral radii among all graphs in $\mathcal{G}_n^{k,\delta}$. 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 $\kappa_g(G)$ 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.