# Further results on the least Q-eigenvalue of a graph with fixed domination number

## TL;DR

This paper investigates the least Q-eigenvalue of connected nonbipartite graphs with fixed order and domination number, providing exact results for specific domination numbers and odd-girth constraints.

## Contribution

It determines the minimum Q-eigenvalue for nonbipartite graphs with given domination number and odd-girth, extending previous spectral graph theory results.

## Key findings

- Minimum Q-eigenvalue for graphs with domination number (n-1)/2 is fully characterized.
- Minimum Q-eigenvalue for graphs with odd-girth ≤ 5 and certain domination numbers is fully characterized.
- Provides new spectral bounds related to domination number and odd-girth in nonbipartite graphs.

## Abstract

In this paper, we proceed on determining the minimum $q_{min}$ among the connected nonbipartite graphs on $n\geq 5$ vertices and with domination number $\frac{n+1}{3}<\gamma\leq \frac{n-1}{2}$. Further results obtained are as follows: $\mathrm{(i)}$ among all nonbipartite connected graph of order $n\geq 5$ and with domination number $\frac{n-1}{2}$, the minimum $q_{min}$ is completely determined; $\mathrm{(ii)}$ among all nonbipartite graphs of order $n\geq 5$, with odd-girth $g_{o}\leq5$ and domination number at least $\frac{n+1}{3}<\gamma\leq \frac{n-2}{2}$, the minimum $q_{min}$ is completely determined.