Extremal distance spectra of graphs and essential connectivity

TL;DR

This paper investigates the extremal properties of graphs and digraphs with fixed essential connectivity, identifying those with the minimum distance spectral radius and characterizing their structure.

Contribution

It determines the extremal graphs and digraphs with minimum distance spectral radius for given essential connectivity and minimum degree, filling a gap in spectral graph theory.

Findings

Identifies extremal n-vertex graphs with minimum distance spectral radius for fixed essential connectivity.

Characterizes extremal graphs with minimum degree and essential connectivity.

Describes extremal strongly connected digraphs with minimum distance spectral radius.

Abstract

A graph is non-trivial if it contains at least one nonloop edge. The essential connectivity of $G$, denoted by $\kappa'(G)$, is the minimum number of vertices of $G$ whose removal produces a disconnected graph with at least two components are non-trivial. In this paper, we determine the $n$-vertex graph of given essential connectivity with minimum distance spectral radius. We also characterize the extremal graphs attaining the minimum distance spectral radius among all connected graphs with fixed essential connectivity and minimum degree. Furthermore, we characterize the extremal digraphs with minimum distance spectral radius among the strongly connected digraphs with given essential connectivity.