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 ʰᵣ (G). Denote by G₍, ^ᵣʰ the set of ʰᵣ-connected graphs of order n with minimum degree. What are the corresponding extremal graphs in G₍, ^ᵣʰ 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 h1. 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 LinJ. Graph Theory (2023) 1--15, as well as Liu, Lu, and TianLinear Algebra Appl. 389 (2004) 139--145, concerning edge connectivity.