On some problems regarding distance-balanced graphs

Abstract

A graph $\Gamma$ is said to be distance-balanced if for any edge $uv$ of $\Gamma$, the number of vertices closer to $u$ than to $v$ is equal to the number of vertices closer to $v$ than to $u$, and it is called nicely distance-balanced if in addition this number is independent of the chosen edge $uv$. A graph $\Gamma$ is said to be strongly distance-balanced if for any edge $uv$ of $\Gamma$ 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 $v$. In this paper we answer an open problem posed by Kutnar and Miklavi\v{c} [European J. Combin. 39 (2014), 57-67] by constructing several infinite families of nonbipartite nicely distance-balanced graphs which are not strongly distance-balanced. We disprove a conjecture regarding characterization of strongly distance-balanced graphs posed by Balakrishnan et al. [European J. Combin. 30 (2009), 1048-1053] by providing infinitely many counterexamples, and answer an open question posed by Kutnar et al. in [Discrete Math. 306 (2006), 1881-1894] regarding existence of semisymmetric distance-balanced graphs which are not strongly distance-balanced by providing an infinite family of such examples. We also show that for a graph $\Gamma$ with $n$ vertices and $m$ edges it can be checked in $O(mn)$ time if $\Gamma$ is strongly-distance balanced and if $\Gamma$ is nicely distance-balanced.