Transactions on Combinatorics (Dec 2016)
On the new extension of distance-balanced graphs
Abstract
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.