## Abstract

In this paper we discuss the chromatic polynomial of a 'bracelet', when the base graph is a complete graph K_{b} and arbitrary links L between the consecutive copies are allowed. If there are n copies of the base graph the resulting graph will be denoted by L_{n}(b). We show that the chromatic polynomial of L_{n}(b) can be written in the form P(L_{n} (b);k)=∑_{ℓ=0}^{b} ∑_{π⊢ℓ}m_{π}(k)tr (N_{L}^{π})^{n}. Here the notation π⊢ℓ means that π is a partition of ℓ, and m_{π}(k) is a polynomial that does not depend on L. The square matrix N_{L}^{π} has size (_{ℓ}^{b}n_{π}, where n_{π} is the degree of the representation R^{π} of Sym_{ℓ} associated with π.We derive an explicit formula for m_{π}(k) and describe a method for calculating the matrices N_{L} ^{π}. Examples are given. Finally, we discuss the application of these results to the problem of locating the chromatic zeros.

Original language | English |
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Pages (from-to) | 147-160 |

Number of pages | 14 |

Journal | European Journal of Combinatorics |

Volume | 25 |

Issue number | 2 |

DOIs | |

State | Published - 1 Feb 2004 |

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics