✦ LIBER ✦

The Matrix of Chromatic Joins

✍ Scribed by W.T. Tutte

Publisher: Elsevier Science
Year: 1993
Tongue: English
Weight: 657 KB
Volume: 57
Category: Article
ISSN: 0095-8956
DOI: 10.1006/jctb.1993.1021

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✦ Synopsis

Matrices (M(n)), one for each positive integer (n), arise in the theory of BirkhoffLewis equations for chromatic polynomials. Students of those equations think it would be helpful to have a formula for the determinant of (M(n)). Finding that determinant is the main object of this paper. It has been conjectured that the determinant is a product of a power of the colour-variable (i) and powers of certain polynomials in (\lambda), those called "Beraha polynomials" by combinatorialists. That would explain the observed occurrences of Beraha polynomials in the solutions of the Birkhoff-Lewis equations for small values of (n). The formula obtained in this paper verifies the conjecture. This paper is closely connected with work done by R. Dahab, D. H. Younger, and the present author on partial solutions of the BirkhoffLewis equations. The first two computed (\operatorname{det} M(n)) up to (n=6), and so made the above-mentioned conjecture seem plausible. 1993 Academic Press, Inc

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