2005/10/18 by K. Kutnar, Klavdija Kutnar, Kutnar, K. +9
Computer Science · Mathematics · #05C07 #05C12 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #math.CO #msc:05C07 #msc:05C12
paper · pdf · doi:10.48550/arxiv.math/0510381
12 pages, 4 figures
arxiv created 2005/10/18 · openalex publication_date 2005/10/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A graph X is said to be \it 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 \it 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. Obviously, being distance--balanced is metrically a weaker condition than being strongly distance--balanced. In this paper, a connection between symmetry properties of graphs and the metric property of being (strongly) distance--balanced is explored. In particular, it is proved that every vertex--transitive graph is strongly distance--balanced. A graph is said to be \em semisymmetric if its automorphism group acts transitively on its edge set, but does not act transitively on its vertex set. An infinite family of semisymmetric graphs, which are not distance--balanced, is constructed. Finally, we give a complete classification of strongly distance--balanced graphs 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).