On the new extension of distance-balanced graphs

Abstract

‎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.