Essential connectivity and spectral radius of graphs

TL;DR

This paper characterizes graphs and digraphs with given essential connectivity and minimum degree that maximize spectral radius, providing exact values and extremal structures.

Contribution

It determines the extremal graphs and digraphs with maximum spectral radius under constraints of essential connectivity and minimum degree.

Findings

Identified graphs with maximum spectral radius in specified classes.

Characterized extremal digraphs with maximum spectral radius.

Provided exact spectral radius values for these extremal digraphs.

Abstract

A graph is trivial if it contains one vertex and no edges. The essential connectivity $\kappa^{\prime}$ of $G$ is defined to be the minimum number of vertices of $G$ whose removal produces a disconnected graph with at least two non-trivial components. Let $\mathcal{A}_n^{\kappa',\delta}$ be the set of graphs of order $n$ with minimum degree $\delta$ and essential connectivity $\kappa'$. In this paper, we determine the graphs attaining the maximum spectral radii among all graphs in $\mathcal{A}_n^{\kappa',\delta}$ and characterize the corresponding extremal graphs. In addition, we also determine the digraphs which achieve the maximum spectral radii among all strongly connected digraphs with given essential connectivity and give the exact values of the spectral radii of these digraphs.