Algebraic methods for chromatic polynomials

N. L. Biggs; M. H. Klin; P. Reinfeld

doi:10.1016/S0195-6698(03)00095-7

Algebraic methods for chromatic polynomials

N. L. Biggs, M. H. Klin, P. Reinfeld

Department of Mathematics

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

5 Scopus citations

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⊢ℓm_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^bn_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
Pages (from-to)	147-160
Number of pages	14
Journal	European Journal of Combinatorics
Volume	25
Issue number	2
DOIs	https://doi.org/10.1016/S0195-6698(03)00095-7
State	Published - 1 Feb 2004

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

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10.1016/S0195-6698(03)00095-7

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title = "Algebraic methods for chromatic polynomials",

abstract = "In this paper we discuss the chromatic polynomial of a 'bracelet', when the base graph is a complete graph Kb 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 Ln(b). We show that the chromatic polynomial of Ln(b) can be written in the form P(Ln (b);k)=∑ℓ=0b ∑π⊢ℓmπ(k)tr (NLπ)n. Here the notation π⊢ℓ means that π is a partition of ℓ, and mπ(k) is a polynomial that does not depend on L. The square matrix NLπ has size (ℓbnπ, where nπ is the degree of the representation Rπ of Symℓ associated with π.We derive an explicit formula for mπ(k) and describe a method for calculating the matrices NL π. Examples are given. Finally, we discuss the application of these results to the problem of locating the chromatic zeros.",

author = "Biggs, {N. L.} and Klin, {M. H.} and P. Reinfeld",

note = "Funding Information: The authors would like to thank P. de la Harpe and R. Bacher for fruitful suggestions, which were communicated by MK to NB and PR at the British Combinatorial Conference in July 2001, and led to the preparation of this paper. We are also grateful to C. Pech for making available his work in the pre-publication version of [14] . The third author is supported by the Engineering and Physical Sciences Research Council.",

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AU - Biggs, N. L.

AU - Klin, M. H.

AU - Reinfeld, P.

N1 - Funding Information: The authors would like to thank P. de la Harpe and R. Bacher for fruitful suggestions, which were communicated by MK to NB and PR at the British Combinatorial Conference in July 2001, and led to the preparation of this paper. We are also grateful to C. Pech for making available his work in the pre-publication version of [14] . The third author is supported by the Engineering and Physical Sciences Research Council.

PY - 2004/2/1

Y1 - 2004/2/1

N2 - In this paper we discuss the chromatic polynomial of a 'bracelet', when the base graph is a complete graph Kb 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 Ln(b). We show that the chromatic polynomial of Ln(b) can be written in the form P(Ln (b);k)=∑ℓ=0b ∑π⊢ℓmπ(k)tr (NLπ)n. Here the notation π⊢ℓ means that π is a partition of ℓ, and mπ(k) is a polynomial that does not depend on L. The square matrix NLπ has size (ℓbnπ, where nπ is the degree of the representation Rπ of Symℓ associated with π.We derive an explicit formula for mπ(k) and describe a method for calculating the matrices NL π. Examples are given. Finally, we discuss the application of these results to the problem of locating the chromatic zeros.

AB - In this paper we discuss the chromatic polynomial of a 'bracelet', when the base graph is a complete graph Kb 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 Ln(b). We show that the chromatic polynomial of Ln(b) can be written in the form P(Ln (b);k)=∑ℓ=0b ∑π⊢ℓmπ(k)tr (NLπ)n. Here the notation π⊢ℓ means that π is a partition of ℓ, and mπ(k) is a polynomial that does not depend on L. The square matrix NLπ has size (ℓbnπ, where nπ is the degree of the representation Rπ of Symℓ associated with π.We derive an explicit formula for mπ(k) and describe a method for calculating the matrices NL π. Examples are given. Finally, we discuss the application of these results to the problem of locating the chromatic zeros.

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VL - 25

SP - 147

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JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

IS - 2

ER -

Algebraic methods for chromatic polynomials

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