on the new extension of distancebalanced graphs

In this paper‎, ‎we initially introduce the concept of n-distance-balanced property which is considered as the generalized concept of distance-balanced property‎. ‎in our consideration‎, ‎we also define the new concept locally regularity in order to find a connection between n-distance-balanced graphs and their lexicographic product‎. ‎furthermore‎, ‎we include a characteristic method which is practicable and can be used to classify all graphs with i-distance-balanced properties for i=2,3 which is also relevant to the concept of total distance‎. ‎moreover‎, ‎we conclude a connection between distance-balanced and 2-distance-balanced graphs‎.