Morteza Faghani - Department of Mathematics‎, ‎Payame Noor University‎, ‎P.O.Box ۹۳۹۵-۳۶۹۷‎, ‎Tehran‎, ‎Iran
Ehsan Pourhadi - School of Mathematics‎, ‎Iran University of Science and Technology‎, ‎Narmak‎, ‎Tehran ۱۶۸۴۶-۱۳۱۱۴‎, ‎Iran
Hassan Kharazi - Department of Mathematics and Statistics‎, ‎Imam Hossein University‎, ‎Tehran‎, ‎Iran

‎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=۲,۳ which is also relevant to the concept of total distance‎. ‎Moreover‎, ‎we conclude a connection between distance-balanced and ۲-distance-balanced graphs‎.

n-distance-balanced property‎, ‎lexicographic product‎, ‎total distance