On some problems regarding distance-balanced graphs

A graph Γ is said to be distance-balanced if for any edge uv of Γ, the number vertices closer u than v equal u, and it called nicely in addition this independent chosen uv. strongly integer k, at distance k from k+1 v. In paper we solve an open problem posed by Kutnar Miklavič (2014) constructing several infinite families nonbipartite graphs which are not distance-balanced. We disprove a conjecture regarding characterization Balakrishnan et al. (2009) providing infinitely many counterexamples, answer question (2006) existence semisymmetric family such examples. also show that with n m edges can checked O(mn) time strongly-distance balanced