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.

中文翻译：

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最大化具有完美匹配的单环图的最小 Q 特征值

让$G\left(G\right)$表示图G的周长。在本文中，我们确定了所有n  = 6 k阶单环图中Q特征值最小的唯一图 $(ķ\ge 8)$搭配完美。对于n  = 6 k  + 2 和n  = 6 k  + 4 的情况，我们证明$G\left(G\right)=3$如果G是具有最大最小Q特征值的图，并提供关于最小Q特征值的尖锐上界的猜想和问题。