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Abstract: 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.
Keywords: Šoltés problem, Wiener index, regular graphs, cubic graphs, Cayley graph, Šoltés vertex
Published in RUP: 10.09.2025; Views: 388; Downloads: 2
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Abstract: In this paper, we present the results of an empirical study examining the achievements of Slovenian elementary school students in arithmetic, with a particular focus on decimal numbers at the levels of basic and conceptual, procedural and problem-solving knowledge. The study aimed to determine whether there are differences or correlations between students' achievements in decimal numbers at these levels of knowledge and whether performance at one level can predict performance at another. Based on an empirical non-experimental study involving 100 Slovenian elementary school students, the findings revealed significant correlations and statistically significant differences between students' achievements at the levels of basic, conceptual, procedural and problem-solving knowledge of decimal numbers. Furthermore, performance at the levels of basic and conceptual, and procedural knowledge were found to predict performance in problem-solving tasks, and vice versa. The study's results indicate that gaps in basic and conceptual or procedural knowledge are reflected in difficulties when solving complex problems, where success often depends on the accuracy of intermediate steps within the solution process.
Keywords: decimal numbers, basic and conceptual knowledge, procedural knowledge, problem-solving knowledge, arithmetic, mathematics
Published in RUP: 11.07.2025; Views: 681; Downloads: 7
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Abstract: In this paper we introduce a problem closely related to the Cage Problem and the Degree Diameter Problem. For integers k ≥ 2, g ≥ 3 and d ≥ 1, we define a (k; g, d)-graph to be a k-regular graph with girth g and diameter d. We denote by n₀(k; g, d) the smallest possible order of such a graph, and, if such a graph exists, we call it a (k; g, d)-cage. In particular, we focus on (k; 5, 4)-graphs. We show that n₀(k; 5, 4) ≥ k² + k + 2 for all k, and report on the determination of all (k; 5, 4)-cages for k = 3, 4 and 5 and of examples with k = 6, and describe some examples of (k; 5, 4)-graphs which prove that n₀(k; 5, 4) ≤ 2k² for infinitely many k.
Keywords: cages, girth, degree-diameter problem
Published in RUP: 10.06.2025; Views: 735; Downloads: 15
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Keywords: sprotni algoritmi, problem pokrivanja košev, konkurenčno razmerje
Published in RUP: 29.08.2024; Views: 1626; Downloads: 34
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Keywords: Extensible Authentication Protocol (EAP), zero-knowledge proofs, authentication, cryptography, key-stretching, passwords authentication, quadratic residuosity problem
Published in RUP: 02.09.2022; Views: 2231; Downloads: 36
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