Maximizing the least Q-eigenvalue of a unicyclic graph with perfect matchings

Abstract

Let $g\left(G\right)$ denote the girth of a graph G. In this paper, we determine the unique graph with the maximum least Q-eigenvalue among all unicyclic graphs of order n = 6k $(k\ge 8)$ with perfect matchings. For the cases when n = 6k + 2 and n = 6k + 4, we prove that $g\left(G\right)=3$ if G is a graph with the maximum least Q-eigenvalue, and provide a conjecture and a problem on the sharp upper bound of the least Q-eigenvalue.

Keywords:

• Signless Laplacian
• least eigenvalue
• unicyclic graph
• upper bound

AMS Classification:

• 05C50

Acknowledgments

We are grateful to the anonymous referees for their careful reading and helpful corrections which result in an improvement of the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Supported by the National Natural Science Foundation of China (Nos. 12071411, 11771376) and the Natural Science Foundation of the Jiangsu Higher Education Institutions. (No. 18KJB110031).