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 1 - 4 / 41 1.Isomorphism checking of I-graphsBoris Horvat, Tomaž Pisanski, Arjana Žitnik, 2012, original scientific articleAbstract: We consider the class of ▫$I$▫-graphs, which is a generalization of the class of the generalized Petersen graphs. We show that two ▫$I$▫-graphs ▫$I(n, j, k)$▫ and ▫$I(n, j_1, k_1)$▫ are isomorphic if and only if there exists an integer ▫$a$▫ relatively prime to $n$ such that either ▫$\{j_1, k_1\} = \{aj \mod n, \; ak \mod n \}$▫ or ▫$\{j_1, k_1\} = \{aj \mod n, \; -ak \mod n\}$▫. This result has an application in the enumeration of non-isomorphic ▫$I$▫-graphs and unit-distance representations of generalized Petersen graphs.Found in: ključnih besedahSummary of found: ...a generalization of the class of the generalized Petersen graphs. We show that two ▫$I$▫-graphs...Keywords: mathematics, graph theory, isomorphism, I-graph, generalized Petersen graphPublished: 15.10.2013; Views: 1560; Downloads: 72 Full text (0,00 KB) 2.Distance-balanced graphs: Symmetry conditionsKlavdija Kutnar, Aleksander Malnič, Dragan Marušič, Štefko Miklavič, 2006, original scientific articleAbstract: A graph ▫$X$▫ is said to be distance-balanced if for any edge ▫$uv$▫ of ▫$X$▫, the number of vertices closer to ▫$u$▫ than to ▫$v$▫ is equal to the number of vertices closer to ▫$v$▫ than to ▫$u$▫. A graph ▫$X$▫ is said to be strongly distance-balanced if for any edge ▫$uv$▫ of ▫$X$▫ and any integer ▫$k$▫, the number of vertices at distance ▫$k$▫ from ▫$u$▫ and at distance ▫$k+1$▫ from ▫$v$▫ is equal to the number of vertices at distance ▫$k+1$▫ from ▫$u$▫ and at distance ▫$k$▫ from ▫$v$▫. Exploring the connection between symmetry properties of graphs and the metric property of being (strongly) distance-balanced is the main theme of this article. That a vertex-transitive graph is necessarily strongly distance-balanced and thus also distance-balanced is an easy observation. With only a slight relaxation of the transitivity condition, the situation changes drastically: there are infinite families of semisymmetric graphs (that is, graphs which are edge-transitive, but not vertex-transitive) which are distance-balanced, but there are also infinite families of semisymmetric graphs which are not distance-balanced. Results on the distance-balanced property in product graphs prove helpful in obtaining these constructions. Finally, a complete classification of strongly distance-balanced graphs is given for the following infinite families of generalized Petersen graphs: GP▫$(n,2)$▫, GP▫$(5k+1,k)$▫, GP▫$(3k 3,k)$▫, and GP▫$(2k+2,k)$▫.Found in: ključnih besedahSummary of found: ...given for the following infinite families of generalized Petersen graphs: GP▫$(n,2)$▫, GP▫$(5k+1,k)$▫, GP▫$(3k 3,k)$▫, and...Keywords: graph theory, graph, distance-balanced graphs, vertex-transitive, semysimmetric, generalized Petersen graphPublished: 15.10.2013; Views: 1889; Downloads: 52 Full text (0,00 KB) 3.The strongly distance-balanced property of the generalized Petersen graphsKlavdija Kutnar, Aleksander Malnič, Dragan Marušič, Štefko Miklavič, 2009, original scientific articleAbstract: A graph ▫$X$▫ is said to be strongly distance-balanced whenever for any edge ▫$uv$▫ of ▫$X$▫ and any positive integer ▫$i$▫, the number of vertices at distance ▫$i$▫ from ▫$u$▫ and at distance ▫$i + 1$▫ from ▫$v$▫ is equal to the number of vertices at distance ▫$i + 1$▫ from ▫$u$▫ and at distance ▫$i$▫ from ▫$v$▫. It is proven that for any integers ▫$k \ge 2$▫ and ▫$n \ge k^2 + 4k + 1$▫, the generalized Petersen graph GP▫$(n, k)$▫ is not strongly distance-balanced.Found in: ključnih besedahSummary of found: ...A graph ▫$X$▫ is said to be strongly distance-balanced...Keywords: graph, strongy distance-balanced, generalized Petersen graphPublished: 15.10.2013; Views: 1487; Downloads: 64 Full text (0,00 KB) 4.Določeni razredi (hiper)grafov in njihove algebraične lastnostiPaweł Petecki, 2016, doctoral dissertationFound in: ključnih besedahSummary of found: ...hypergraph, hamiltonian cycle, decomposition, double generalized Petersen graph, automorphism group, vertex-transitive, sign graph,...Keywords: hypergraph, hamiltonian cycle, decomposition, double generalized Petersen graph, automorphism group, vertex-transitive, sign graph, L-eigenvalue, lollipop graphPublished: 09.08.2016; Views: 1423; Downloads: 3 Full text (0,00 KB)
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