Extremal distance spectral radius of graphs with $h$-extra $r$-component connectivity

Abstract

For two integers $r\ge 2$ and $h\ge 0$, the $h$-extra $r$-component connectivity of a graph $G$, denoted by $c{\kappa }_r^h$, is defined as the minimum number of vertices whose removal produces a disconnected graph with at least $r$ components, where each component contains at least $h+1$ vertices. Let ${\mathcal{G}}_{n,\delta }^{c{\kappa }_r^h}$ represent the set of graphs of order $n$ with minimum degree $\delta$ and $h$-extra $r$-component connectivity $c{\kappa }_r^h$. Hu, Lin, and Zhang [\textit{Discrete Math.} \textbf{345} (2025) 114621] investigated the case when $h=0$ within ${\mathcal{G}}_{n,\delta }^{c{\kappa }_r^h}$, and characterized the corresponding extremal graphs that minimize the distance spectral radius. In this paper, we further explore the relevant extremal graphs in ${\mathcal{G}}_{n,\delta }^{c{\kappa }_r^h}$ for $h\ge 1$.