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Letdenote 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  = 6 kwith perfect matchings. For the cases when  n  = 6 k  + 2 and  n  = 6 k  + 4, we prove thatif  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.

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