Maximum spectral radius of graphs with given connectivity and minimum degree

TL;DR

This paper determines the maximum spectral radius of graphs with given connectivity and minimum degree, identifying the unique extremal graph structure.

Contribution

It extends previous work by characterizing the maximum spectral radius for graphs with both connectivity and minimum degree constraints.

Findings

Maximum spectral radius achieved at a specific graph structure

Unique extremal graph identified for given parameters

Generalizes previous results on spectral radius bounds

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 $K_{k}+(K_{\delta-k+1}\cup K_{n-\delta-1})$.