scholarly journals Galois groups of chromatic polynomials

2012 ◽  
Vol 15 ◽  
pp. 281-307 ◽  
Author(s):  
Kerri Morgan
Keyword(s):  
Galois Group ◽  
Systematic Study ◽  
Statistical Physics ◽  
Infinite Family ◽  
Chromatic Polynomial ◽  
Galois Groups ◽  
Chromatic Polynomials ◽  
Algebraic Properties

AbstractThe chromatic polynomialP(G,λ) gives the number of ways a graphGcan be properly coloured in at mostλcolours. This polynomial has been extensively studied in both combinatorics and statistical physics, but there has been little work on its algebraic properties. This paper reports a systematic study of the Galois groups of chromatic polynomials. We give a summary of the Galois groups of all chromatic polynomials of strongly non-clique-separable graphs of order at most 10 and all chromatic polynomials of non-clique-separableθ-graphs of order at most 19. Most of these chromatic polynomials have symmetric Galois groups. We give five infinite families of graphs: one of these families has chromatic polynomials with a dihedral Galois group and two of these families have chromatic polynomials with cyclic Galois groups. This includes the first known infinite family of graphs that have chromatic polynomials with the cyclic Galois group of order 3.

scholarly journals Certificates of Factorisation for a Class of Triangle-Free Graphs

10.37236/164 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Kerri Morgan ◽  
Graham Farr
Keyword(s):  
Infinite Family ◽  
Chromatic Polynomial ◽  
Chromatic Polynomials ◽  
Odd Cycle ◽  
Separable Graph

The chromatic polynomial $P(G,\lambda)$ gives the number of $\lambda$-colourings of a graph. If $P(G,\lambda)=P(H_{1},\lambda)P(H_{2},\lambda)/P(K_{r},\lambda)$, then the graph $G$ is said to have a chromatic factorisation with chromatic factors $H_{1}$ and $H_{2}$. It is known that the chromatic polynomial of any clique-separable graph has a chromatic factorisation. In this paper we construct an infinite family of graphs that have chromatic factorisations, but have chromatic polynomials that are not the chromatic polynomial of any clique-separable graph. A certificate of factorisation, that is, a sequence of rewritings based on identities for the chromatic polynomial, is given that explains the chromatic factorisations of graphs from this family. We show that the graphs in this infinite family are the only graphs that have a chromatic factorisation satisfying this certificate and having the odd cycle $C_{2n+1}$, $n\geq2$, as a chromatic factor.

scholarly journals Classification of Algebraic Properties of Chromatic Polynomials

2013 ◽  
Vol 5 (3) ◽  
pp. 469-477 ◽  
Author(s):  
R. V. N. S. Rao ◽  
J. V. Rao
Keyword(s):  
Chromatic Polynomial ◽  
Möbius Function ◽  
Chromatic Polynomials ◽  
Inversion Theorem ◽  
Mobius Function ◽  
Möbius Inversion ◽  
Algebraic Properties

This manuscript attempts to introduce the concept of chromatic polynomials of total graphs using Mobius inversion theorem. In fact it studies various algebraic properties of chromatic polynomial using Mobius inversion theorem. Keywords: Bond lattice; Chromatic polynomial; Mobius function; Poset.  © 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.  doi: http://dx.doi.org/10.3329/jsr.v5i3.11634 J. Sci. Res. 5 (3), 469-477 (2013)

scholarly journals On the Galois group of three classes of trinomials

2021 ◽  
Vol 7 (1) ◽  
pp. 212-224
Author(s):  
Lingfeng Ao ◽  
◽  
Shuanglin Fei ◽  
Shaofang Hong
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Prime Number ◽  
Galois Groups

<abstract><p>Let $ n\ge 8 $ be an integer and let $ p $ be a prime number satisfying $ \frac{n}{2} &lt; p &lt; n-2 $. In this paper, we prove that the Galois groups of the trinomials</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ T_{n, p, k}(x): = x^n+n^kp^{(n-1-p)k}x^p+n^kp^{nk}, $\end{document} </tex-math></disp-formula></p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ S_{n, p}(x): = x^n+p^{n(n-1-p)}n^px^p+n^pp^{n^2} $\end{document} </tex-math></disp-formula></p> <p>and</p> <p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ E_{n, p}(x): = x^n+pnx^{n-p}+pn^2 $\end{document} </tex-math></disp-formula></p> <p>are the full symmetric group $ S_n $ under several conditions. This extends the Cohen-Movahhedi-Salinier theorem on the irreducible trinomials $ f(x) = x^n+ax^s+b $ with integral coefficients.</p></abstract>

scholarly journals Algebraic Properties of Chromatic Roots

10.37236/6578 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Kerri Morgan
Keyword(s):  
Algebraic Integer ◽  
Tutte Polynomial ◽  
Chromatic Polynomial ◽  
Galois Groups ◽  
Algebraic Numbers ◽  
Isaac Newton ◽  
Isaac Newton Institute ◽  
Chromatic Root ◽  
The Subject ◽  
Chromatic Roots

A chromatic root is a root of the chromatic polynomial of a graph.  Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008.  The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

scholarly journals Calculating Galois groups of third-order linear differential equations with parameters

2018 ◽  
Vol 20 (04) ◽  
pp. 1750038
Author(s):  
Andrei Minchenko ◽  
Alexey Ovchinnikov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Galois Group ◽  
Galois Groups ◽  
Linear Differential Equations ◽  
Unipotent Radical ◽  
Third Order ◽  
Differential Galois Group ◽  
Bessel Differential Equation ◽  
Linear Differential

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation.

Endomorphisms of Two Dimensional Jacobians and Related Finite Algebras

2012 ◽  
Vol 55 (1) ◽  
pp. 38-47
Author(s):  
William Butske
Keyword(s):  
Galois Group ◽  
Galois Groups ◽  
Two Dimensional ◽  
Finite Algebras ◽  
Genus 2

AbstractZarhin proves that if C is the curve y2 = f (x) where Galℚ(f(x)) = Sn or An, then . In seeking to examine his result in the genus g = 2 case supposing other Galois groups, we calculate for a genus 2 curve where f (x) is irreducible. In particular, we show that unless the Galois group is S5 or A5, the Galois group does not determine .

scholarly journals On Galois groups of class two extensions over the rational number field

1979 ◽  
Vol 75 ◽  
pp. 121-131 ◽  
Author(s):  
Susumu Shirai
Keyword(s):  
Nilpotent Group ◽  
Galois Group ◽  
Number Field ◽  
Direct Product ◽  
Rational Number ◽  
Nilpotency Class ◽  
Abelian Extension ◽  
Galois Groups ◽  
Classification Theory ◽  
Ray Class Field

Let Q be the rational number field, K/Q be a maximal Abelian extension whose degree is some power of a prime l, and let f(K) be the conductor of K/Q; if l = 2, let K be complex, and if in addition f(K) ≡ 0 (mod 2), let f(K) ≡ 0 (mod 16). Denote by (K) the Geschlechtermodul of K over Q and by K̂ the maximal central l-extension of K/Q contained in the ray class field mod (K) of K. A. Fröhlich [1, Theorem 4] completely determined the Galois group of K̂ over Q in purely rational terms. The proof is based on [1, Theorem 3], though he did not write the proof in the case f(K) ≡ 0 (mod 16). Moreover he gave a classification theory of all class two extensions over Q whose degree is a power of l. Hence we know the set of fields of nilpotency class two over Q, because a finite nilpotent group is a direct product of all its Sylow subgroups. But the theory becomes cumbersome, and it is desirable to reconstruct a more elementary one.

scholarly journals Computing the Galois group of a polynomial over a p-adic field

2020 ◽  
Vol 16 (08) ◽  
pp. 1767-1801 ◽  
Author(s):  
Christopher Doris
Keyword(s):  
Galois Group ◽  
Galois Groups ◽  
Naive Algorithm

We present a family of algorithms for computing the Galois group of a polynomial defined over a p-adic field. Apart from the “naive” algorithm, these are the first general algorithms for this task. As an application, we compute the Galois groups of all totally ramified extensions of [Formula: see text] of degrees 18, 20 and 22, tables of which are available online.

scholarly journals GALOIS GROUPS AND AN OBSTRUCTION TO PRINCIPAL GRAPHS OF SUBFACTORS

2007 ◽  
Vol 18 (02) ◽  
pp. 191-202 ◽  
Author(s):  
MARTA ASAEDA
Keyword(s):  
Recent Result ◽  
Galois Group ◽  
Numerical Computation ◽  
Galois Groups ◽  
Jones Index ◽  
Principal Graph ◽  
New Type ◽  
Index Value

The Galois group of the minimal polymonal of a Jones index value gives a new type of obstruction to a principal graph, thanks to a recent result of Etingof, Nikshych, and Ostrik. We show that the sequence of the graphs given by Haagerup as candidates of principal graphs of subfactors, are not realized as principal graphs for 7 < n ≤ 27 using GAP program. We further utilize Mathematica to extend the statement to 27 < n ≤ 55. We conjecture that none of the graphs are principal graphs for all n > 7, and give an evidence using Mathematica for smaller graphs among them for n > 55. The problem for the case n = 7 remains open, however, it is highly likely that it would be realized as a principal graph, thanks to numerical computation by Ikeda.

On 2-Groups as Galois Groups

1995 ◽  
Vol 47 (6) ◽  
pp. 1253-1273 ◽  
Author(s):  
Arne Ledet
Keyword(s):  
Galois Group ◽  
Direct Product ◽  
Galois Extension ◽  
Quaternion Algebra ◽  
Abelian Groups ◽  
Galois Groups ◽  
Embedding Problem ◽  
Brauer Group ◽  
Characteristic 2 ◽  
Cyclic Kernel

AbstractLet L/K be a finite Galois extension in characteristic ≠ 2, and consider a non-split Galois theoretical embedding problem over L/K with cyclic kernel of order 2. In this paper, we prove that if the Galois group of L/K is the direct product of two subgroups, the obstruction to solving the embedding problem can be expressed as the product of the obstructions to related embedding problems over the corresponding subextensions of L/K and certain quaternion algebra factors in the Brauer group of K. In connection with this, the obstructions to realising non-abelian groups of order 8 and 16 as Galois groups over fields of characteristic ≠ 2 are calculated, and these obstructions are used to consider automatic realisations between groups of order 4, 8 and 16.

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