### 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 CAY_{d}(n) be the number of isomorphism classes of d-valent Cayley graphs of order at most n. We show that log(CAY_{d}(n))∈Θ(d(logn)^{2}), as n→∞. We also obtain some stronger results in the case d=3.

Original language | English |
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Pages (from-to) | 221-240 |

Number of pages | 20 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 122 |

DOIs | |

Publication status | Published - 1 Jan 2017 |

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### Cite this

*Journal of Combinatorial Theory. Series B*,

*122*, 221-240. https://doi.org/10.1016/j.jctb.2016.06.002

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*Journal of Combinatorial Theory. Series B*, vol. 122, pp. 221-240. https://doi.org/10.1016/j.jctb.2016.06.002

**Asymptotic enumeration of vertex-transitive graphs of fixed valency.** / Potočnik, Primož; Spiga, Pablo; Verret, Gabriel.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Asymptotic enumeration of vertex-transitive graphs of fixed valency

AU - Potočnik, Primož

AU - Spiga, Pablo

AU - Verret, Gabriel

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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.

AB - 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.

KW - 3-Valent

KW - Cayley

KW - Cubic

KW - Enumeration

KW - GRR

KW - Vertex-transitive

UR - http://www.scopus.com/inward/record.url?scp=84997161550&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2016.06.002

DO - 10.1016/j.jctb.2016.06.002

M3 - Article

VL - 122

SP - 221

EP - 240

JO - Journal of combinatorial Theory Series B

JF - Journal of combinatorial Theory Series B

SN - 0095-8956

ER -