Uniform Solt\'es' hypergraphs and Solt\'es' weighted graphs
Abstract
A Solt\'es' hypergraph is a hypergraph for which the removal of any of its vertices does not change its total distance. We prove that every uniform Solt\'es' hypergraph has order at least 10, there exist uniform Solt\'es' hypergraphs for almost every order or uniformity, and there exist a non-regular uniform Solt\'es' hypergraph. By also providing infinitely many weighted Solt\'es' graphs, we conclude that Solt\'es' problem can be answered positively for the most natural generalisations of graphs.
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