on the new extension of distance-balanced graphs

‎In this paper‎, ‎we initially introduce the concept of n 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 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 i-distance-balanced properties for i=2,3 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.