Uniform Šoltés' hypergraphs first occur at order 10, exist for every order n >= 10 and most uniformities k >= 4, a non-regular 9-uniform example exists, and infinitely many weighted Šoltés' graphs exist.
The maximum Wiener index of a uniform hypergraph
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abstract
The Wiener index of a (hyper)graph is calculated by summing up the distances between all pairs of vertices. We determine the maximum possible Wiener index of a connected $n$-vertex $k$-uniform hypergraph and characterize for every~$n$ all hypergraphs attaining the maximum Wiener index.
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Uniform \v{S}olt\'es' hypergraphs and \v{S}olt\'es' weighted graphs
Uniform Šoltés' hypergraphs first occur at order 10, exist for every order n >= 10 and most uniformities k >= 4, a non-regular 9-uniform example exists, and infinitely many weighted Šoltés' graphs exist.