| Title: | Distance-balanced graphs: Symmetry conditions |
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| Authors: | ID Kutnar, Klavdija (Author)
ID Malnič, Aleksander (Author)
ID Marušič, Dragan (Author)
ID Miklavič, Štefko (Author) |
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| Files: |  http://dx.doi.org/10.1016/j.disc.2006.03.066
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| Language: | English |
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| Work type: | Not categorized |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | IAM - Andrej Marušič Institute
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| Abstract: | 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▫$(2k+2,k)$▫. |
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| Keywords: | graph theory, graph, distance-balanced graphs, vertex-transitive, semysimmetric, generalized Petersen graph |
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| Year of publishing: | 2006 |
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| Number of pages: | str. 1881-1894 |
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| Numbering: | Vol. 306, iss. 16 |
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| PID: | 20.500.12556/RUP-835  |
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| ISSN: | 0012-365X |
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| UDC: | 519.17 |
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| COBISS.SI-ID: | 1714391 |
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| Publication date in RUP: | 15.10.2013 |
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| Views: | 5698 |
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| Downloads: | 91 |
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