## Abstract

Apart from the role the clustering coefficient plays in the definition of the small-world phenomena, it also has great relevance for practical problems involving networked dynamical systems. To study the impact of the clustering coefficient on dynamical processes taking place on networks, some authors have focused on the construction of graphs with tunable clustering coefficients. These constructions are usually realized through a stochastic process, either by growing a network through the preferential attachment procedure, or by applying a random rewiring process. In contrast, we consider here several families of static graphs whose clustering coefficients can be determined explicitly. The basis for these families is formed by the k-regular graphs on N nodes, that belong to the family of so-called circulant graphs denoted by C_{N,k}. We show that the expression for the clustering coefficient of C_{N,k} reported in literature, only holds for sufficiently large N. Next, we consider three generalizations of the circulant graphs, either by adding some pendant links to C_{N,k}, or by connecting, in two different ways, an additional node to some nodes of C_{N,k}. For all three generalizations, we derive explicit expressions for the clustering coefficient. Finally, we construct a family of pairs of generalized circulant graphs, with the same number of nodes and links, but with different clustering coefficients.

Original language | English |
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Article number | 126088 |

Pages (from-to) | 1-13 |

Number of pages | 13 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 578 |

DOIs | |

Publication status | Published - 2021 |

### Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-careOtherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

## Keywords

- Circulant graphs
- Clustered networks
- Complex networks
- Consensus dynamics