Connectivity and Minimal Distance Spectral Radius of Graphs

TL;DR

This paper investigates how the distance spectral radius of graphs changes under edge grafting and identifies graphs with minimal spectral radius given a fixed number of cut vertices or edges.

Contribution

It introduces new insights into the behavior of the distance spectral radius under graph perturbations and characterizes extremal graphs with minimal spectral radius for specific cut conditions.

Findings

Distance spectral radius decreases or increases predictably with grafting.

Graphs with k cut vertices or edges that minimize spectral radius are characterized.

Provides methods to identify extremal graphs for given cut constraints.

Abstract

In this paper, we study how the distance spectral radius behaves when the graph is perturbed by grafting edges. As applications, we also determine the graph with $k$ cut vertices (respectively, $k$ cut edges) with the minimal distance spectral radius.