Extremal spectral radius in h-extra r-component (edge)-connected graphs

For two integers r≥ 2 and h≥ 0, a subset S⊆ V is called an h-extra r-component cut of G such that there are at least r connected components in $G\!-\!S$ and each component has at least $h+1$ vertices. The minimum size of any h-extra r-component cut of G, if it exists, is the h-extra r-component connectivity of G, denoted as κʰᵣ(G). Denote by Gn,δκᵣʰ the set of κʰᵣ-connected graphs of order n with minimum degree δ. What are the corresponding extremal graphs in Gn,δκᵣʰ with the maximum spectral radius for r≥ 2 and h≥ 0? Fan, Gu and Lin [l-connectivity, l-edge-connectivity and spectral radius of graphs, arXiv:2309.05247] give the answer to r≥ 2 and $h=0$. In this paper, we provide a solution for r≥ 2 and h≥1. Moreover, we also investigate analogous problems for the edge version. Our results can imply some previous results, specifically those of Lu and Lin [J. Discrete Algorithms 31 (2015) 113--119] in connectivity, and for the work of Fan, Gu, and Lin[J. Graph Theory (2023) 1--15], as well as Liu, Lu, and Tian[Linear Algebra Appl. 389 (2004) 139--145], concerning edge connectivity.