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Chromatic polynomials and representations of the symmetric group

Norman Biggs
2002 Linear Algebra and its Applications  
In the present paper the levels are explained by using a version of the sieve principle, and it is shown that the terms at level correspond to the irreducible representations of the symmetric group Sym  ...  Previous calculations for b = 2 and b = 3 suggest that the chromatic polynomial contains terms that occur in 'levels'.  ...  symmetric group Sym .  ... 
doi:10.1016/s0024-3795(01)00610-3 fatcat:kgo42ulp6ncpzkloxdlsnchsdm

Galois groups of chromatic polynomials

Kerri Morgan
2012 LMS Journal of Computation and Mathematics  
Most of these chromatic polynomials have symmetric Galois groups.  ...  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  ...  I thank Graham Farr for suggesting this topic and for giving regular feedback on drafts of this paper. I thank John Cannon for his support and for providing us with a copy of Magma.  ... 
doi:10.1112/s1461157012001052 fatcat:zvgosqoaqrbhndtkipmx2n6zq4

Algebraic properties of chromatic roots [article]

Peter J. Cameron, Kerri Morgan
2016 arXiv   pre-print
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.  ...  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 are grateful to the Institute for the excellent facilities and opportunities for interaction which it provided.  ... 
arXiv:1610.00424v1 fatcat:zopghzaazzdjpks5m2fzfdgkom

Graph coloring-related properties of (generating functions of) Hodge-Deligne polynomials [article]

Soohyun Park
2022 arXiv   pre-print
Finally, we give an application of methods used to symmetries of Hodge-Deligne polynomials of varieties and their configuration spaces and their relation to chromatic symmetric polynomials.  ...  Motivated by a connection between the topology of (generalized) configuration spaces and chromatic polynomials, we show that generating functions of Hodge-Deligne polynomials of quasiprojective varieties  ...  ACKNOWLEDGEMENTS I am very thankful to my advisor Benson Farb for helpful discussions and encouragement throughout the project.  ... 
arXiv:2203.11930v1 fatcat:ppcuzq32onfsdfrjywbbxy73ve

A categorification of the chromatic symmetric polynomial

Radmila Sazdanović, Martha Yip
2015 Discrete Mathematics & Theoretical Computer Science  
International audience The Stanley chromatic polynomial of a graph $G$ is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties.  ...  We also obtain analogues of several familiar properties of the chromatic symmetric polynomials in terms of homology.  ...  Acknowledgements The former author would like to thank the Simons Foundation for its support via the AMS Travel and Simons Collaboration grants.  ... 
doi:10.46298/dmtcs.2527 fatcat:ubn3lesmzreahi7oqgctojwj6e

A categorification of the chromatic symmetric function

Radmila Sazdanovic, Martha Yip
2018 Journal of combinatorial theory. Series A  
The Stanley chromatic symmetric polynomial of a graph G is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties.  ...  We also obtain analogues of several familiar properties of the chromatic symmetric polynomials in terms of homology. Résumé.  ...  Acknowledgements The former author would like to thank the Simons Foundation for its support via the AMS Travel and Simons Collaboration grants.  ... 
doi:10.1016/j.jcta.2017.08.014 fatcat:cttvz2ndu5d5pfhs4xowot4eam

Chromatic symmetric function of graphs from Borcherds algebras [article]

G. Arunkumar
2021 arXiv   pre-print
The absolute value of the linear coefficient of the chromatic polynomial of G is known as the chromatic discriminant of G.  ...  We prove that the chromatic symmetric function of G can be recovered from the Weyl denominator identity of 𝔤 and this gives a Lie theoretic proof of Stanley's expression for chromatic symmetric function  ...  Since, the chromatic symmetric functions are generalization of chromatic polynomials it is natural ask for the connection between chromatic symmetric functions and Borcherds algebras.  ... 
arXiv:1908.08198v2 fatcat:ukroh7oi4jf2nct7bsorwoob5y

Algebraic invariants arising from the chromatic polynomials of theta graphs

Daniel Delbourgo, Kerri Morgan
2014 The Australasian Journal of Combinatorics  
We give a complete description of the Galois group, discriminant and ramification indices for the chromatic polynomials of theta graphs with three consecutive path lengths.  ...  This paper investigates some algebraic properties of the chromatic polynomials of theta graphs, i.e. graphs which have three internally disjoint paths sharing the same two distinct end vertices.  ...  They also thank the Australian Research Council for their support under an ARC-DP110100957 grant.  ... 
dblp:journals/ajc/DelbourgoM14 fatcat:27nx7r7h7jg6hiddiockdtowpq

Coloring With a Limited Paintbox

Stephanie van Willigenburg
2022 Notices of the American Mathematical Society  
If you retort with some very simple case which makes me out a stupid animal, I think I must do as the Sphynx did. 1 Hamilton wrote back on  ...  On October 23, 1852 Francis Guthrie asked his brother Frederick, a student at University College London, England, to ask his professor, Augustus De Morgan, whether four colors sufficed to color the countries  ...  stories made writing and researching this article all the more chromatic.  ... 
doi:10.1090/noti2490 fatcat:ia2ybtyhlzhxzazz62426mbzoq

Transplanting Trees: Chromatic Symmetric Function Results through the Group Algebra of S_n [article]

Angèle M. Foley, Joshua Kazdan, Larissa Kröll, Sofía Martínez Alberga, Oleksii Melnyk, Alexander Tenenbaum
2022 arXiv   pre-print
One of the major outstanding conjectures in the study of chromatic symmetric functions (CSF's) states that trees are uniquely determined by their CSF's.  ...  Additionally, we prove that a "parent function" of the CSF defined in the group ring of S_n can uniquely determine trees, providing further support for Stanley's conjecture.  ...  The author Angèle Foley was supported by an NSERC Discovery Grant and the authors Joshua Kazdan and Sofía Martínez Alberga are supported by the NSF GRFP: Award # 1650114.  ... 
arXiv:2112.09937v2 fatcat:hlnlx3k4gzg6bm2grtv7ulq7cq

From poset topology to 𝑞-Eulerian polynomials to Stanley's chromatic symmetric functions [chapter]

John Shareshian, Michelle L. Wachs
2016 The Mathematical Legacy of Richard P. Stanley  
In recent years we have worked on a project involving poset topology, various analogues of Eulerian polynomials, and a refinement of Richard Stanley's chromatic symmetric function.  ...  Here we discuss how Stanley's ideas and results have influenced and inspired our own work.  ...  as the chromatic symmetric function of G n .  ... 
doi:10.1090//mbk/100/18 fatcat:gg5lpzddhzcxhcvmptke7eqhdy

From Poset Topology to q-Eulerian Polynomials to Stanley's Chromatic Symmetric Functions [article]

John Shareshian, Michelle L. Wachs
2015 arXiv   pre-print
In recent years we have worked on a project involving poset topology, various analogues of Eulerian polynomials, and a refinement of Richard Stanley's chromatic symmetric function.  ...  Here we discuss how Stanley's ideas and results have influenced and inspired our own work.  ...  Since the hard Lefschetz map on H * (H(m)) commutes with the action of the symmetric group, the conjecture implies that X inc(P (m)) (x; t) is Schur-unimodal, which in turn implies that Conjecture 4.7  ... 
arXiv:1505.03530v1 fatcat:23qf77nybvenvhg6mawenjhdda

Orbital Chromatic and Flow Roots

PETER J. CAMERON, K. K. KAYIBI
2006 Combinatorics, probability & computing  
The chromatic polynomial P Γ (x) of a graph Γ is a polynomial whose value at the positive integer k is the number of proper kcolourings of Γ.  ...  Our hypotheses include parity conditions on the elements of G and also some special types of graphs and groups. We also look at orbital flow roots.  ...  Let Γ be the null graph on n vertices, and G the symmetric group S n .  ... 
doi:10.1017/s0963548306008200 fatcat:m3g3gyq2o5ejbl47zo7xzecxwu

The Chromatic Polynomial of a Digraph [article]

Winfried Hochstättler, Johanna Wiehe
2022 arXiv   pre-print
This decomposition will confirm the equality of our chromatic polynomial of a digraph and the chromatic polynomial of the underlying undirected graph in the case of symmetric digraphs.  ...  Furthermore we will decompose our NL-coflow polynomial, which becomes the chromatic polynomial of a digraph by multiplication with the number of colors to the number of components, examining the special  ...  Probably, the chromatic polynomial of a graph is better known than the flow polynomial.  ... 
arXiv:1911.09547v3 fatcat:jhmi3sxkjrhqtcgr6mtufluvx4

Combinatorics of multivariate chromatic polynomials for rooted graphs [article]

Nicholas A. Loehr, Gregory S. Warrington
2022 arXiv   pre-print
Richard Stanley defined the chromatic symmetric function X_G of a graph G and conjectured that trees T and U are isomorphic if and only if X_T=X_U.  ...  The first is that our polynomials satisfy the analogue of Stanley's conjecture: two rooted trees are isomorphic as rooted graphs if and only if their rooted chromatic polynomials are equal.  ...  Acknowledgments The authors thank Sergei Chmutov, Peter McNamara, Rosa Orellana, John Shareshian, Stephanie van Willigenburg, and Victor Wang for supplying references and other helpful comments regarding  ... 
arXiv:2206.05392v2 fatcat:eh4i5u76i5h4tf52yjosp3576a
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