Distance-balanced graphs: Symmetry conditions

Klavdija Kutnar (Author), Aleksander Malnič (Author), Dragan Marušič (Author), Štefko Miklavič (Author)

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;

Data

Language:	English
Year of publishing:	2006
Typology:	1.01 - Original Scientific Article
Organization:	UP - University of Primorska
UDC:	519.17
COBISS:	1714391
ISSN:	0012-365X
Views:	4078
Downloads:	89
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary keywords:	teorija grafov;razdaljno uravnoteženi grafi;polsimetrični grafi;posplošeni Petersenov graf;tranzitivni grafi;
Type (COBISS):	Not categorized
Pages:	str. 1881-1894
Volume:	ǂVol. ǂ306
Issue:	ǂiss. ǂ16
Chronology:	2006
ID:	1471767