Distance-balanced graphs: Symmetry conditions

A graph X is said to be distance-balanced   if for any edge uvuv of X, the number of vertices closer to u   than to vv is equal to the number of vertices closer to vv than to u. A graph X is said to be strongly distance-balanced   if for any edge uvuv of X and any integer k, the number of vertices at distance k from u   and at distance k+1k+1 from vv is equal to the number of vertices at distance k+1k+1 from u and at distance k   from vv. 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(n,2), GP(5k+1,k)GP(5k+1,k), GP(3k±3,k)GP(3k±3,k), and GP(2k+2,k)GP(2k+2,k).