Asymptotic enumeration of vertex-transitive graphs of fixed valency

Primož Potočnik; Pablo Spiga; Gabriel Verret

doi:10.1016/j.jctb.2016.06.002

Asymptotic enumeration of vertex-transitive graphs of fixed valency

Primož Potočnik, Pablo Spiga, Gabriel Verret

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

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(log⁡n)²), as n→∞. We also obtain some stronger results in the case d=3.

Original language	English
Pages (from-to)	221-240
Number of pages	20
Journal	Journal of Combinatorial Theory. Series B
Volume	122
DOIs	https://doi.org/10.1016/j.jctb.2016.06.002
Publication status	Published - 1 Jan 2017

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title = "Asymptotic enumeration of vertex-transitive graphs of fixed valency",

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(log⁡n)2), as n→∞. We also obtain some stronger results in the case d=3.",

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AU - Potočnik, Primož

AU - Spiga, Pablo

AU - Verret, Gabriel

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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(log⁡n)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(log⁡n)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

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VL - 122

SP - 221

EP - 240

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

ER -

Asymptotic enumeration of vertex-transitive graphs of fixed valency

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