On the generalized distance spectral radius of graphs

TL;DR

This paper investigates the generalized distance spectral radius of graphs, characterizing extremal graphs with minimum spectral radius under various constraints, thus extending known spectral graph theory results.

Contribution

It provides the first characterization of graphs with minimum generalized distance spectral radius for fixed chromatic number, trees, and unicyclic graphs.

Findings

Identifies the unique graph with minimum spectral radius for fixed chromatic number.

Determines graphs with minimum spectral radius among n-vertex trees.

Determines graphs with minimum spectral radius among unicyclic graphs.

Abstract

The generalized distance spectral radius of a connected graph $G$ is the spectral radius of the generalized distance matrix of $G$, defined by $$D_\alpha(G)=\alpha Tr(G)+(1-\alpha)D(G), \;\;0\le\alpha \le 1,$$ where $D(G)$ and $Tr(G)$ denote the distance matrix and diagonal matrix of the vertex transmissions of $G$, respectively. This paper characterizes the unique graph with minimum generalized distance spectral radius among the connected graphs with fixed chromatic number, which answers a question about the generalized distance spectral radius in spectral extremal theories. In addition, we also determine graphs with minimum generalized distance spectral radius among the $n$-vertex trees and unicyclic graphs, respectively. These results generalize some known results about distance spectral radius and distance signless Laplacian spectral radius of graphs.