scholarly journals Chromatic Roots and Minor-Closed Families of Graphs

2016 ◽  
Vol 30 (3) ◽  
pp. 1883-1897
Author(s):  
Thomas Perrett
Keyword(s):  
Chromatic Roots

scholarly journals Chromatic roots—some observations and conjectures

1980 ◽  
Vol 29 (2) ◽  
pp. 161-167 ◽  
Author(s):  
E.J. Farrell
Keyword(s):  
Chromatic Roots

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 On chromatic roots of large subdivisions of graphs

2002 ◽  
Vol 242 (1-3) ◽  
pp. 17-30 ◽  
Author(s):  
J.I. Brown ◽  
C.A. Hickman
Keyword(s):  
Chromatic Roots

scholarly journals Subdivisions and Chromatic Roots

1999 ◽  
Vol 76 (2) ◽  
pp. 201-204 ◽  
Author(s):  
Jason I. Brown
Keyword(s):  
Chromatic Roots

scholarly journals Chromatic roots at 2 and the Beraha number B10

2020 ◽  
Vol 95 (3) ◽  
pp. 445-456
Author(s):  
Daniel J. Harvey ◽  
Gordon F. Royle
Keyword(s):  
Chromatic Roots

scholarly journals Chromatic zeros and the golden ratio

2009 ◽  
Vol 3 (1) ◽  
pp. 120-122 ◽  
Author(s):  
Saeid Alikhani ◽  
Yee-Hock Peng
Keyword(s):  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Chromatic Polynomial ◽  
Chromatic Roots

In this note, we investigate ?n , where ?=1+2?5 is the golden ratio as chromatic roots. Using some properties of Fibonacci numbers, we prove that ? n (n ? N), cannot be roots of any chromatic polynomial.

scholarly journals On the Chromatic Roots of Generalized Theta Graphs

2001 ◽  
Vol 83 (2) ◽  
pp. 272-297 ◽  
Author(s):  
Jason I. Brown ◽  
Carl Hickman ◽  
Alan D. Sokal ◽  
David G. Wagner
Keyword(s):  
Chromatic Roots

scholarly journals Density of Chromatic Roots in Minor-Closed Graph Families

2018 ◽  
Vol 27 (6) ◽  
pp. 988-998 ◽  
Author(s):  
THOMAS J. PERRETT ◽  
CARSTEN THOMASSEN
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Classical Result ◽  
Golden Ratio ◽  
Chromatic Polynomial ◽  
Closed Graph ◽  
Small Interval ◽  
Chromatic Polynomials ◽  
Graph Families ◽  
Chromatic Roots

We prove that the roots of the chromatic polynomials of planar graphs are dense in the interval between 32/27 and 4, except possibly in a small interval around τ + 2 where τ is the golden ratio. This interval arises due to a classical result of Tutte, which states that the chromatic polynomial of every planar graph takes a positive value at τ + 2. Our results lead us to conjecture that τ + 2 is the only such number less than 4.

scholarly journals Benjamini-Schramm convergence and the distribution of chromatic roots for sparse graphs

COMBINATORICA ◽  
2014 ◽  
Vol 35 (2) ◽  
pp. 127-151 ◽  
Author(s):  
Miklós Abért ◽  
Tamás Hubai
Keyword(s):  
Sparse Graphs ◽  
Chromatic Roots
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