Abstract
A graph Γ is distance-balanced if for each pair u,v of adjacent vertices of Γ the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. Apart from the interest in these graphs from the graph theoretical point of view they have applications in other areas of research, for instance in mathematical chemistry and communication networks, and have thus been studied from various different points of view in the literature.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.
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