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For two integers $r\geq 2$ and $h\geq 0$, the \emph{$h$-extra $r$-component connectivity} $\kappa^h_r(G)$ of a graph $G$ is defined to be the minimum size of a subset of vertices whose removal disconnects $G$, and there are at least $r$ connected components in $G\!-\!S$ and each component has at least $h+1$ vertices. Denote by $\mathcal{G}_{n,\delta}^{\kappa_r^h}$ the set of graphs with $h$-extra $r$-component connectivity $\kappa^h_r(G)$ and minimum degree $\delta$. The following problem concerning spectral radius was proposed by Brualdi and Solheid [On the spectral radius of complementary acyclic matrices of zeros and one, SIAM J. Algebra Discrete Methods 7 (1986) 265-272]: Given a set of graphs $\mathscr{S}$, find an upper bound for the spectral radius of graphs in $\mathscr{S}$ and characterize the graphs in which the maximal spectral radius is attained. We study this question for $\mathscr{S}=\mathcal{G}_{n,\delta}^{\kappa_r^h}$ where $r\geq 2$ and $h\geq 0$. Fan, Gu and Lin [$l$-connectivity, $l$-edge-connectivity and spectral radius of graphs, \emph{arXiv}:2309.05247] give the answer to $r\geq 2$ and $h=0$. In this paper, we solve this problem completely for $r\geq 2$ and $h

Author

Wang, Yu; Li, Dan; Lin, Huiqiu

Published

2024

Language

EN