Distance-balanced graphs: symmetry conditions

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: $\text{GP}(n,2)$, $\text{GP}(5k+1,k)$, $\text{GP}(3k\pm 3,k)$, and $\text{GP}(2k+2,k)$.