AI 中文总结
本文研究拉普拉斯量子游走的图不变量，确定了完全图$K_n$、星型图$S_n$等图的惰性排序，还发现顶点数相同时更不平衡的双星图惰性更强。
AI 中文摘要
量子游走的平均混合矩阵的迹衡量了该游走的“惰性”：迹越高，游走者长期返回起点的概率越大。本文开发了用于研究拉普拉斯量子游走产生的该图不变量的工具。已知完全图$K_n$是$n$个顶点上最惰性的连通图。利用我们的方法，我们证明了星型图$S_n$是$n$个顶点上第二惰性的连通图（因此是$n$个顶点上最惰性的树），完全多部图$K_{n-2,1,1}$是$n$个顶点上第三惰性的连通图，双星图$DS(n-3,1)$是$n$个顶点上第二惰性的树。我们还证明，在顶点数相同的情况下，更不平衡的双星图惰性更强。
英文摘要
The trace of the average mixing matrix of a quantum walk measures the "laziness" of the walk: the higher the trace, the more likely that the walker returns home in the long run. In this paper, we develop tools to study this graph invariant arising from Laplacian quantum walks. It is known that the complete graph $K_n$ is the laziest connected graph on $n$ vertices. Using our machinery, we show that the star $S_n$ is the second laziest connected graph on $n$ vertices (and hence the laziest tree on $n$ vertices), the complete multipartite graph $K_{n-2,1,1}$ is the third laziest connected graph on $n$ vertices, and the double star $DS(n-3,1)$ is the second laziest tree on $n$ vertices. We also show that on the same number of vertices, more unbalanced double stars are lazier.
