Distance Sequences In Locally Infinite Vertex-Transitive Digraphs

We prove that the out-distance sequence {f+(k)} of a vertex-transitive digraph of finite or infinite degree satisfies f+(k+1)≤f+(k)2 for k≥1, where f+(k) denotes the number of vertices at directed distance k from a given vertex. As a corollary, we prove that for a connected vertex-transitive undirected graph of infinite degree d, we have f(k)=d for all k, 1≤k<diam(G). This answers a question by L. Babai.

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Correspondence to Wesley Pegden.

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Pegden, W. Distance Sequences In Locally Infinite Vertex-Transitive Digraphs. Combinatorica 26, 577–585 (2006). https://doi.org/10.1007/s00493-006-0033-y

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Mathematics Subject Classification (2000):

  • 05C12