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

Let g (G) 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 ≥ 8) with perfect matchings. For the cases when n = 6k + 2 and n = 6k + 4, we prove that g (G) = 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. [ABSTRACT FROM AUTHOR]