1. Distance-regular Cayley graphs over ℤpˢ ⊕ ℤpXiongfeng Zhan, Lu Lu, Xueyi Huang, 2025, original scientific article Abstract: In 2007, Miklavič and Potočnik proposed the problem of characterizing distance-regular Cayley graphs, which can be viewed as an extension of the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. Let p be an odd prime. In this paper, all distance-regular Cayley graphs over ℤps ⊕ ℤp are identified. It is shown that every such graph is isomorphic to a complete graph, a complete multipartite graph, or the line graph of a transversal design TD(r, p) with 2 ≤ r ≤ p − 1.
Keywords: distance-regular graph, Cayley graph, Schur ring, Fourier transformation, transversal design
Published in RUP: 21.10.2025; Views: 341; Downloads: 1
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2. The Gray graph is a unit-distance graphLeah Berman, Gábor Gévay, Tomaž Pisanski, 2025, original scientific article Abstract: In this note we give a construction proving that the Gray graph, which is the smallestcubic semisymmetric graph, is a unit-distance graph.
Keywords: polycirculant, unit-distance graph, Gray graph, ADAM graph, generalized Petersen graph
Published in RUP: 10.09.2025; Views: 489; Downloads: 2
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3. Tetravalent distance magic graphs of small order and an infinite family of examplesKsenija Rozman, Primož Šparl, 2025, original scientific article Abstract: A graph of order ▫$n$▫ is distance magic if it admits a bijective labeling of its vertices with integers from ▫$1$▫ to ▫$n$▫ such that each vertex has the same sum of the labels of its neighbors. This paper contributes to the long term project of characterizing all tetravalent distance magic graphs. With the help of a computer we find that out of almost nine million connected tetravalent graphs up to order 16 only nine are distance magic. In fact, besides the six well known wreath graphs there are only three other examples, one of each of the orders 12, 14 and 16. We introduce a generalization of wreath graphs, the so-called quasi wreath graphs, and classify all distance magic graphs among them. This way we obtain infinitely many new tetravalent distance magic graphs. Moreover, the two non-wreath graphs of orders 12 and 14 are quasi wreath graphs while the one of order 16 can be obtained from a quasi wreath graph of order 14 using a simple construction due to Kovář, Fronček and Kovářová.
Keywords: distance magic, tetravalent, quasi wreath graph
Published in RUP: 10.09.2025; Views: 356; Downloads: 5
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4. The distance function on Coxeter-like graphs and self-dual codesMarko Orel, Draženka Višnjić, 2025, original scientific article Keywords: Coxeter graph, invertible symmetric matrices, binary field, rank, distance in graphs, alternate matrices, self-dual codes
Published in RUP: 30.05.2025; Views: 808; Downloads: 18
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5. Selected topics on Wiener indexMartin Knor, Riste Škrekovski, Aleksandra Tepeh, 2024, original scientific article Keywords: graph distance, Wiener index, average distance, topological index, molecular descriptor, chemical graph theory
Published in RUP: 26.05.2025; Views: 751; Downloads: 7
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8. Mathematical aspects of Wiener indexMartin Knor, Riste Škrekovski, Aleksandra Tepeh, 2016, original scientific article Abstract: The Wiener index (i.e., the total distance or the transmission number), defined as the sum of distances between all unordered pairs of vertices in a graph, is one of the most popular molecular descriptors. In this article we summarize some results, conjectures and problems on this molecular descriptor, with emphasis on works we were involved in.
Keywords: Wiener index, total distance, topological index, molecular descriptor, chemical graph theory
Published in RUP: 03.01.2022; Views: 3835; Downloads: 49
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9. On minimal forbidden subgraphs for the class of EDM-graphsGašper Jaklič, Jolanda Modic, 2015, original scientific article Abstract: In this paper, a relation between graph distance matrices and Euclidean distance matrices (EDM) is considered. Graphs, for which the distance matrix is not an EDM (NEDM-graphs), are studied. All simple connected non-isomorphic graphs on ▫$n \le 8$▫ nodes are analysed and a characterization of the smallest NEDM-graphs, i.e., the minimal forbidden subgraphs, is given. It is proven that bipartite graphs and some subdivisions of the smallest NEDM-graphs are NEDM-graphs, too.
Keywords: graph theory, graph, Euclidean distance matrix, distance, eigenvalue
Published in RUP: 31.12.2021; Views: 2331; Downloads: 21
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