Nicely distance-balanced graphs

A nonempty graph ΓΓ is called nicely distance-balanced  , whenever there exists a positive integer γΓγΓ, such that for any two adjacent vertices u,vu,v of ΓΓ there are exactly γΓγΓ vertices of ΓΓ which are closer to uu than to vv, and exactly γΓγΓ vertices of ΓΓ which are closer to vv than to uu. The aim of this paper is to introduce the notion of nicely distance-balanced graphs, to provide examples of such graphs, to discuss relations between these graphs and distance-balanced and strongly distance-balanced graphs, as well as to prove some other results regarding these graphs.