A unified approach to the spectral radius, connectivity and edge-connectivity of graphs

For two integers $r\ge 2$ and $h\ge 0$, the h-extra r-component connectivity ${\kappa }_r^h\left(G\right)$ of a graph G is defined as the minimum size of a subset S of vertices whose removal disconnects G, such that there are at least r connected components in $G-S$ and each component has at least $h+1$ vertices. Denote by $G_{n,\delta }^{{\kappa }_r^h}$ the set of n-vertex graphs with h-extra r-component connectivity ${\kappa }_r^h$ and minimum degree δ. The following problem concerning spectral radius was proposed by Brualdi and Solheid (1986) [2]: Given a set of graphs $S$, find an upper bound for the spectral radius of graphs in $S$ and characterize the graphs in which the maximum spectral radius is attained. We study this question for $S=G_{n,\delta }^{{\kappa }_r^h}$ where $r\ge 2$ and $h\ge 0$. Fan, Gu and Lin (2024) [7] answered the question for $r\ge 2$ and $h=0$. In this paper, we solve this problem completely for $r\ge 2$ and $h\ge 1$. Moreover, we also investigate analogous problems for the edge version. This implies some previous results in connectivity and edge-connectivity.