SUMMARY
Throughout this paper, we present a new strong property of graph so-called nicely n-distance-balanced which is notably stronger than the concept of n-distance-balanced recently given by the authors. We also initially introduce a newgraph invariant which modies Szeged index and is suitable to study n-distance-balanced graphs. Looking for the graphs extremal with respect to the modiedSzeged index it turns out the n-distance-balanced graphs with odd integer n arethe only bipartite graphs which can maximize the modied Szeged index and thisalso disproves a conjecture proposed by Khalifeh et al. [Khalifeh M.H.,Youse-Azari H., Ashra A.R., Wagner S.G.: Some new results on distance-based graphinvariants, European J. Combin. 30 (2009) 1149-1163]. Furthermore, we gathersome facts concerning with the nicely n-distance-balanced graphs generated by somewell-known graph products. To enlighten the reader some examples are provided.Moreover, a conjecture and a problem are presented within the results of this article.