Disproof of a conjecture on the minimum spectral radius and the domination number

TL;DR

This paper disproves a conjecture regarding the minimizer graph with the smallest spectral radius among connected graphs with a given domination number, and fully characterizes the minimizer for odd n.

Contribution

It refutes a previous conjecture and precisely identifies the unique minimizer graph for odd n in the specified class.

Findings

Disproved the conjecture on the minimizer graph for odd n.

Determined the unique minimizer graph among G_{n,⌊n/2⌋} for odd n.

Confirmed the minimizer graph is a tree for all cases.

Abstract

Let $G_{n,\gamma}$ be the set of all connected graphs on $n$ vertices with domination number $\gamma$. A graph is called a minimizer graph if it attains the minimum spectral radius among $G_{n,\gamma}$. Very recently, Liu, Li and Xie [Linear Algebra and its Applications 673 (2023) 233--258] proved that the minimizer graph over all graphs in $\mathbb{G}_{n,\gamma}$ must be a tree. Moreover, they determined the minimizer graph among $G_{n,\lfloor\frac{n}{2}\rfloor}$ for even $n$, and posed the conjecture on the minimizer graph among $G_{n,\lfloor\frac{n}{2}\rfloor}$ for odd $n$. In this paper, we disprove the conjecture and completely determine the unique minimizer graph among $G_{n,\lfloor\frac{n}{2}\rfloor}$ for odd $n$.