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

Abstract: For two integers $r\geq 2$ and $h\geq 0$, 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 $h\geq 1$.