Unexpectedly, a symmetry on unlabeled graphs

TL;DR

This paper demonstrates a surprising symmetry in the distribution of two graph parameters on unlabeled connected graphs, using species theory, and highlights the novelty of this equidistribution phenomenon.

Contribution

The paper introduces a new symmetric distribution of two graph parameters on unlabeled graphs and employs species theory for its proof, a novel approach in this context.

Findings

Proves a symmetric distribution of parameters on unlabeled graphs

Uses species theory for enumeration and proof

Highlights the first known natural equidistribution of this kind

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.