ON THE NEW EXTENSION OF DISTANCE-BALANCED 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 dene the new concept locally regularity in order to nd 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.