We prove a sharp upper bound on the number of shortest cycles contained inside any connected graph in terms of its number of vertices, girth, and maximal degree. Equality holds only for Moore graphs, which gives a new characterization of these graphs. In the case of regular graphs, our result improves an inequality of Teo and Koh. We also show that a subsequence of the Ramanujan graphs of Lubotzky-Phillips-Sarnak have super-linear kissing numbers.

中文翻译：

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正则图的接吻数

我们证明了任何连通图中包含的最短环数的尖锐上限，就其顶点数、周长和最大度数而言。等式仅适用于摩尔图，这为这些图提供了新的表征。在正则图的情况下，我们的结果改进了 Teo 和 Koh 的不等式。我们还表明 Lubotzky-Phillips-Sarnak 的拉马努金图的子序列具有超线性接吻数。