Explicit formulas for chromatic polynomials of some series-parallel graphs

E.Yu. Lerner, S.A. Mukhamedjanova

Kazan Federal University, Kazan, 420008 Russia

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For citation: Lerner E.Yu., Mukhamedjanova S.A. Explicit formulas for chromatic polynomials of some series-parallel graphs. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 339–349. 

Для цитирования: Lerner E.Yu., Mukhamedjanova S.A. Explicit formulas for chromatic polynomials of some series-parallel graphs // Учен. зап. Казан. ун-та. Сер. Физ.-матем. науки. – 2018. – Т. 160, кн. 2. – С. 339–349. 

Abstract

The main goal of our paper is to present explicit formulas for chromatic polynomials of some planar series-parallel graphs (sp-graphs). The necklace-graph considered in this paper is the simplest non-trivial sp-graph. We have provided the explicit formula for calculating the chromatic polynomial of common sp-graphs. In addition, we have presented the explicit formulas for calculating chromatic polynomials of the ring of the necklace graph and the necklace of the necklace graph. Chromatic polynomials of the necklace graph and the ring of the necklace graph have been initially obtained by transition to the dual graph and the subsequent using of the flow polynomial. We have also used the technique of finite Fourier transformations. The use of the partition function of the Potts model is a more general way to evaluate chromatic polynomials. In this method, we have used the parallel- and series-reduction identities that were introduced by A. Sokal. We have developed this idea and introduced the transformation of the necklace-graph reduction. Using this transformation makes it easier to calculate chromatic polynomials for the necklace-graph, the ring of the necklace graph, as well as allows to calculate the chromatic polynomial of the necklace of the necklace graph.

Keywords: chromatical polynomial, partition function of Potts model, Tutte polynomial, Fourier transform, series-parallel graph, necklace graph

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Received

November 27, 2017

Lerner Eduard Yulievich, Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Analysis Data and Operations Research

Kazan Federal University

ul. Kremlevskaya, 18, Kazan, 420008 Russia

E-mail:  eduard.lerner@gmail.com

Mukhamedjanova Sofya Alfisovna, Postgraduate Student of the Department of Mathematical Statistics and Theory of Probability

Kazan Federal University

ul. Kremlevskaya, 18, Kazan, 420008 Russia

E-mail:  sofya_92@mail.ru

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