ℓ-distance-balanced graphs

In this paper we study a very natural generalization of the concept of distance-balancedness, introduced by Frelih (2014). Let ℓ denote a positive integer. A connected graph Γ of diameter at least ℓ is said to be ℓ-distance-balanced whenever for any pair of vertices u,v of Γ at distance ℓ, the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. We obtain some general results on ℓ-distance-balanced graphs and provide various examples. We study those of diameter at most 3 in more detail and investigate the ℓ-distance-balancedness property of cubic graphs. In particular, we analyze this property for the generalized Petersen graphs.