On regular graphs with Šoltés vertices

english

2025

volume 25, issue 2, article p2.01

Cayley graph   cubic graphs   regular graphs   Šoltés problem   Šoltés vertex   Wiener index

Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije
(Obvezni izvod spletne publikacije)

Let ?$W(G)$? be the Wiener index of a graph ?$G$?. We say that a vertex ?$v \in V(G)$? is a Šoltés vertex in ?$G$? if ?$W(G - v) = W(G)$?, i.e. the Wiener index does not change if the vertex ?$v$? is removed. In 1991, Šoltés posed the problem of identifying all connected graphs ?$G$? with the property that all vertices of ?$G$? are Šoltés vertices. The only such graph known to this day is ?$C_{11}$?. As the original problem appears to be too challenging, several relaxations were studied: one may look for graphs with at least ?$k$? Šoltés vertices; or one may look for ?$\alpha$?-Šoltés graphs, i.e. graphs where the ratio between the number of Šoltés vertices and the order of the graph is at least ?$\alpha$?. Note that the original problem is, in fact, to find all ?$1$?-Šoltés graphs. We intuitively believe that every ?$1$?-Šoltés graph 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 Šoltés vertices. In this paper, we present several partial results. For every ?$r\ge 1$? we describe a construction of an infinite family of cubic ?$2$?-connected graphs with at least ?$2^r$? Šoltés vertices. Moreover, we report that a computer search on publicly available collections of vertex-transitive graphs did not reveal any ?$1$?-Šoltés graph. We are only able to provide examples of large ?$\frac{1}{3}$?-Šoltés graphs that are obtained by truncating certain cubic vertex-transitive graphs. This leads us to believe that no ?$1$?-Šoltés graph other than ?$C_{11}$? exists.

20.02.2026