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arXiv · 2505.03665

Unexpectedly, a symmetry on unlabeled graphs

Florian Fürnsinn·Moritz Gangl·Martin Rubey
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Abstract

We exhibit the joint symmetric distribution of the following two parameters on the set of unlabeled, simple, connected graphs with $n$ vertices. The first parameter is the maximal number of leaves attached to a vertex. The second parameter is the size of the largest set of vertices sharing the same closed neighborhood minus $1$. Apparently, this is the first example of a natural, non-trivial equidistribution of graph parameters on unlabeled connected graphs on a fixed set of vertices. Our proof is enumerative, using the theory of species. Exhibiting an explicit bijection interchanging the two parameters remains an open problem.

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Florian Fürnsinn, Moritz Gangl, Martin Rubey. 2025-05-06. Unexpectedly, a symmetry on unlabeled graphs. https://arxiv.org/abs/2505.03665

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