On regular graphs with Šoltés vertices

Bašić, Nino (avtor), Knor, Martin (avtor), Škrekovski, Riste (avtor)

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.

• Jezik: Angleški jezik
• Ključne besede: Šoltés problem, Wiener index, regular graphs, cubic graphs, Cayley graph, Šoltés vertex
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Različica publikacije: Objavljena publikacija
• Datum objave: 01.01.2025
• Leto izida: 2025
• Št. strani: 20 str.
• Številčenje: Vol. 25, no. 2, article no. P2.01
• PID: 20.500.12556/RUL-168564
• UDK: 519.17
• ISSN pri članku: 1855-3966
• DOI: 10.26493/1855-3974.3085.3ea
• COBISS.SI-ID: 232776195
• Datum objave v RUL: 17.04.2025

Gradivo je del revije

• Naslov: Ars mathematica contemporanea
• Založnik: Društvo matematikov, fizikov in astronomov, Društvo matematikov, fizikov in astronomov, Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije
• ISSN: 1855-3966
• COBISS.SI-ID: 239049984

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.