The minimum spectral radius of graphs with a given domination number

TL;DR

This paper investigates the graphs with the smallest spectral radius among connected graphs with a fixed number of vertices and domination number, characterizing the minimizers for specific domination values.

Contribution

It proves that the minimizer graphs are trees and characterizes all such minimizers for certain domination numbers.

Findings

Minimizer graphs are trees.

Complete characterization for specific domination numbers.

Spectral radius minimized by particular tree structures.

Abstract

Let $\mathbb{G}_{n,\gamma}$ be the set of simple and connected graphs on $n$ vertices and with domination number $\gamma$. The graph with minimum spectral radius among $\mathbb{G}_{n,\gamma}$ is called the minimizer graph. In this paper, we first prove that the minimizer graph of $\mathbb{G}_{n,\gamma}$ must be a tree. Moreover, for $\gamma\in\{1,2,3,\lceil\frac{n}{3}\rceil,\lfloor\frac{n}{2}\rfloor\}$, we characterize all minimizer graphs in $\mathbb{G}_{n,\gamma}$.