Geodesic
On 2-distance-balanced graphs
Ars Mathematica Contemporanea, Tome 15 (2018) no. 1, pp. 81-95.

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Let n denote a positive integer. A graph Γ of diameter at least n is said to be n-distance-balanced whenever for any pair of vertices u, v of Γ at distance n, the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. In this article we consider n = 2 (e.g. we consider 2-distance-balanced graphs). We show that there exist 2-distance-balanced graphs that are not 1-distance-balanced (e.g. distance-balanced). We characterize all connected 2-distance-balanced graphs that are not 2-connected. We also characterize 2-distance-balanced graphs that can be obtained as cartesian product or lexicographic product of two graphs.
DOI : 10.26493/1855-3974.1382.dee
Keywords: n-distance-balanced graph, cartesian product, lexicographic product 
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Boštjan Frelih; Štefko Miklavič. On 2-distance-balanced graphs. Ars Mathematica Contemporanea, Tome 15 (2018) no. 1, pp. 81-95. doi : 10.26493/1855-3974.1382.dee. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.1382.dee/

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