Maximum spectral radius of graphs with given connectivity and minimum degree

Abstract

Shiu, Chan and Chang [On the spectral radius of graphs with connectivity at most $k$, J. Math. Chem., 46 (2009), 340-346] studied the spectral radius of graphs of order $n$ with $\kappa(G) \leq k$ and showed that among those graphs, the maximum spectral radius is obtained uniquely at $K_k^n$, which is the graph obtained by joining $k$ edges from $k$ vertices of $K_{n-1}$ to an isolated vertex. In this paper, we study the spectral radius of graphs of order $n$ with $\kappa(G)\leq k$ and minimum degree $\delta(G)\geq k $. We show that among those graphs, the maximum spectral radius is obtained uniquely at $K_{k}+(K_{\delta-k+1}\cup K_{n-\delta-1})$.