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Title:Distance-balanced graphs: Symmetry conditions
Authors:Kutnar, Klavdija (Author)
Malnič, Aleksander (Author)
Marušič, Dragan (Author)
Miklavič, Štefko (Author)
Files:URL http://dx.doi.org/10.1016/j.disc.2006.03.066
 
Language:English
Work type:Not categorized
Tipology:1.01 - Original Scientific Article
Organization:IAM - Andrej Marušič Institute
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▫$(2k+2,k)$▫.
Keywords:graph theory, graph, distance-balanced graphs, vertex-transitive, semysimmetric, generalized Petersen graph
Year of publishing:2006
Number of pages:str. 1881-1894
Numbering:Vol. 306, iss. 16
ISSN:0012-365X
UDC:519.17
COBISS_ID:1714391 Link is opened in a new window
Views:1900
Downloads:53
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Secondary language

Language:English
Keywords:teorija grafov, razdaljno uravnoteženi grafi, polsimetrični grafi, posplošeni Petersenov graf, tranzitivni grafi

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