  For every pair of vertices u and v with d(u; v) = n, Wun G v denotes the set
of all vertices of G that are closer to u than to v. In this paper, we
introduce quasi-({\lambda}; n)-distance-balanced graphs and then study some
properties of these graphs and present a formula to construct such graphs for
arbitrarily diameter d. For n = 1, this class of graphs contains the
quasi-{\lambda}-DB graphs recently introduced by Abedi et al.
[Quasi-{\lambda}-distance-balanced graphs, Discrete Appl. Math. 227 (2017)
21{28]. Moreover, we will take a look at the problems arisen by Abedi et al.
Some problems and conjecture are involved.
