Nicely distance-balanced graphs

A nonempty graph Γ is called nicely distance-balanced, whenever there exists a positive integer γ Γ , such that for any two adjacent vertices u , v of Γ there are exactly γ Γ vertices of Γ which are closer to u than to v , and exactly γ Γ vertices of Γ which are closer to v than to u . 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.