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Let $W(G)$ be the Wiener index of a graph $G$. We say that a vertex $v \inV(G)$ is a \v{S}olt\'es vertex in $G$ if $W(G - v) = W(G)$, i.e. the Wienerindex does not change if the vertex $v$ is removed. In 1991, \v{S}olt\'es posedthe problem of identifying all connected graphs $G$ with the property that allvertices of $G$ are \v{S}olt\'es vertices. The only such graph known to thisday is $C_{11}$. As the original problem appears to be too challenging, severalrelaxations were studied: one may look for graphs with at least $k$ \v{S}oltesvertices; or one may look for $\alpha$-\v{S}olt\'es graphs, i.e. graphs wherethe ratio between the number of \v{S}olt\'es vertices and the order of thegraph is at least $\alpha$. Note that the original problem is, in fact, to findall $1$-\v{S}olt\'es graphs. We intuitively believe that every $1$-\v{S}olt\'esgraph has to be regular and has to possess a high degree of symmetry.Therefore, we are interested in regular graphs that contain one or more\v{S}olt\'es vertices. In this paper, we present several partial results. Wedescribe a construction that gives rise to an infinite family of cubic$2$-connected graphs with two \v{S}olt\'es vertices. Moreover, we report that acomputer search on publicly available collections of vertex-transitive graphsdid not reveal any $1$-\v{S}olt\'es graph. We are only able to provide examplesof large $\frac{1}{3}$-\v{S}olt\'es graphs that are obtained by truncatingcertain cubic vertex-transitive graphs. This leads us to believe that no$1$-\v{S}olt\'es graph other than $C_{11}$ exists.

On regular graphs with \v{S}olt\'es vertices