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.
Towards the essence of \v{S}olt\'es' problem
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abstract
We explore the question asking for graphs $G$ for which the total distance decreases, possibly by a fixed constant $k$, upon the removal of any of its vertices. We obtain results leading to intuition and doubts for the \v{S}olt\'es' problem ($k=0$) and its conjectures.
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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.