Let G be a group and let S be an inverse-closed and identity-free generating set of G. The Cayley graph Cay(G,S) has vertex-set G and two vertices u and v are adjacent if and only if uv−1∈S. Let CAYd(n) be the number of isomorphism classes of d-valent Cayley graphs of order at most n. We show that log(CAYd(n))∈Θ(d(logn)2), as n→∞. We also obtain some stronger results in the case d=3.
Potočnik, P., Spiga, P., Verret, G. (2017). Asymptotic enumeration of vertex-transitive graphs of fixed valency. JOURNAL OF COMBINATORIAL THEORY, 122, 221-240 [10.1016/j.jctb.2016.06.002].
Asymptotic enumeration of vertex-transitive graphs of fixed valency
Spiga, P
;
2017
Abstract
Let G be a group and let S be an inverse-closed and identity-free generating set of G. The Cayley graph Cay(G,S) has vertex-set G and two vertices u and v are adjacent if and only if uv−1∈S. Let CAYd(n) be the number of isomorphism classes of d-valent Cayley graphs of order at most n. We show that log(CAYd(n))∈Θ(d(logn)2), as n→∞. We also obtain some stronger results in the case d=3.
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