{"?xml":{"@version":"1.0"},"edm:RDF":{"@xmlns:dc":"http://purl.org/dc/elements/1.1/","@xmlns:edm":"http://www.europeana.eu/schemas/edm/","@xmlns:wgs84_pos":"http://www.w3.org/2003/01/geo/wgs84_pos","@xmlns:foaf":"http://xmlns.com/foaf/0.1/","@xmlns:rdaGr2":"http://rdvocab.info/ElementsGr2","@xmlns:oai":"http://www.openarchives.org/OAI/2.0/","@xmlns:owl":"http://www.w3.org/2002/07/owl#","@xmlns:rdf":"http://www.w3.org/1999/02/22-rdf-syntax-ns#","@xmlns:ore":"http://www.openarchives.org/ore/terms/","@xmlns:skos":"http://www.w3.org/2004/02/skos/core#","@xmlns:dcterms":"http://purl.org/dc/terms/","edm:WebResource":[{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-5PQXBDBR/3fd7d685-965c-44d9-9332-0bb3e3f1272f/PDF","dcterms:extent":"456 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-5PQXBDBR/0f832405-1bfe-4033-8b86-dd772dde9109/TEXT","dcterms:extent":"50 KB"}],"edm:TimeSpan":{"@rdf:about":"2008-2025","edm:begin":{"@xml:lang":"en","#text":"2008"},"edm:end":{"@xml:lang":"en","#text":"2025"}},"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:doc-5PQXBDBR","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2025","dc:creator":["Bašić, Nino","Knor, Martin","Škrekovski, Riste"],"dc:format":[{"@xml:lang":"sl","#text":"številka:2, article  p2.01"},{"@xml:lang":"sl","#text":"20 str."},{"@xml:lang":"sl","#text":"letnik:25"}],"dc:identifier":["DOI:10.26493/1855-3974.3085.3ea","ISSN:1855-3966","COBISSID_HOST:232776195","URN:URN:NBN:SI:doc-5PQXBDBR"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije"},"dc:subject":[{"@xml:lang":"en","#text":"Cayley graph"},{"@xml:lang":"en","#text":"cubic graphs"},{"@xml:lang":"en","#text":"regular graphs"},{"@xml:lang":"en","#text":"Šoltés problem"},{"@xml:lang":"en","#text":"Šoltés vertex"},{"@xml:lang":"en","#text":"Wiener index"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"On regular graphs with Šoltés vertices|"},"dc:description":{"@xml:lang":"sl","#text":"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"},"edm:type":"TEXT","dc:type":[{"@xml:lang":"sl","#text":"znanstveno časopisje"},{"@xml:lang":"en","#text":"journals"},{"@rdf:resource":"http://www.wikidata.org/entity/Q361785"}]},"ore:Aggregation":{"@rdf:about":"http://www.dlib.si/?URN=URN:NBN:SI:doc-5PQXBDBR","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-5PQXBDBR"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-5PQXBDBR/3fd7d685-965c-44d9-9332-0bb3e3f1272f/PDF"},"edm:rights":{"@rdf:resource":"http://creativecommons.org/licenses/by/4.0/"},"edm:provider":"Slovenian National E-content Aggregator","edm:intermediateProvider":{"@xml:lang":"en","#text":"National and University Library of Slovenia"},"edm:dataProvider":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije"},"edm:object":{"@rdf:resource":"http://www.dlib.si/streamdb/URN:NBN:SI:doc-5PQXBDBR/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-5PQXBDBR"}}}}