Distance-balanced graphs: Symmetry conditions

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. 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 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). IntroductionLet X be a graph with diameter d, and let V (X) and E(X) denote the vertex set and the edge set of X, respectively. For u, v ∈ V (X), we let d(u, v) denote the minimal path-length distance between u and v. We say that X is distance-balanced ifholds for an arbitrary pair of adjacent vertices u and v of X. These graphs were, at least implicitly, first studied by Handa [8] who considered distance-balanced partial cubes. The term itself, however, is due to Jerebic, Klavžar and Rall [11] who studied distance-balanced graphs in the framework of various kinds of graph products.

Nicely distance-balanced graphs

“…As GP(n, 1) is vertex-transitive, it is SDB by [13,Corollary 2.2]. To see that GP(n, 1) is not NDB, consider the edge u 0 v 0 .…”

“…A regular edge-transitive graph which is not vertex-transitive is called semisymmetric. In [13] it is proved that vertex-transitive graphs are not only DB; they are also SDB. But there are infinitely many semisymmetric graphs which are not DB, as well as there are infinitely many semisymmetric graphs which are DB.…”

“…For recent results on DB and SDB graphs see [1,4,8,[13][14][15][16]. The aim of this paper is to introduce the notion of NDB graphs, to provide examples of such graphs, to discuss relations between NDB, DB and SDB property, as well as to prove some other results regarding these graphs.…”

On 2-distance-balanced graphs

“…family of distance-balanced graphs is very rich (for instance, every distance-regular graph as well as every vertex-transitive graph has this property [13]). In the literature these graphs were studied from various purely graph-theoretic aspects such as symmetry [13], connectivity [9,16] or complexity aspects of algorithms related to such graphs [6], to name just a few.…”

“…In the literature these graphs were studied from various purely graph-theoretic aspects such as symmetry [13], connectivity [9,16] or complexity aspects of algorithms related to such graphs [6], to name just a few. However, it turns out that these graphs have applications in other areas, such as mathematical chemistry (see for instance [3,11,12]) and communication networks (see for instance [3]).…”