# On the new extension of distance-balanced graphs

Document Type : Research Paper

Authors

1 Department of Mathematics‎, ‎Payame Noor University‎, ‎P.O.Box 9395-3697‎, ‎Tehran‎, ‎Iran

2 School of Mathematics‎, ‎Iran University of Science and Technology‎, ‎Narmak‎, ‎Tehran 16846-13114‎, ‎Iran

3 Department of Mathematics and Statistics‎, ‎Imam Hossein University‎, ‎Tehran‎, ‎Iran

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.

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