Bounds For The Real Zeros of Chromatic Polynomials

2008 ◽  
Vol 17 (6) ◽  
pp. 749-759 ◽  
Author(s):  
F. M. DONG ◽  
K. M. KOH
Keyword(s):  
Maximum Degree ◽  
Short Proof ◽  
Chromatic Polynomial ◽  
Chromatic Polynomials ◽  
The Real ◽  
Real Zeros ◽  
Complex Zeros ◽  
Special Case

Sokal in 2001 proved that the complex zeros of the chromatic polynomialPG(q) of any graphGlie in the disc |q| < 7.963907Δ, where Δ is the maximum degree ofG. This result answered a question posed by Brenti, Royle and Wagner in 1994 and hence proved a conjecture proposed by Biggs, Damerell and Sands in 1972. Borgs gave a short proof of Sokal's result. Fernández and Procacci recently improved Sokal's result to |q| < 6.91Δ. In this paper, we shall show that all real zeros ofPG(q) are in the interval [0,5.664Δ). For the special case that Δ = 3, all real zeros ofPG(q) are in the interval [0,4.765Δ).

scholarly journals Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions

2001 ◽  
Vol 10 (1) ◽  
pp. 41-77 ◽  
Author(s):  
ALAN D. SOKAL
Keyword(s):  
Partition Function ◽  
Simple Proof ◽  
Potts Model ◽  
Maximum Degree ◽  
Partition Functions ◽  
Polymer Model ◽  
Chromatic Polynomials ◽  
Reliability Polynomial ◽  
Complex Zeros ◽  
Special Case

We show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree [les ] r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc [mid ]q[mid ] < C(r). Furthermore, C(r) [les ] 7.963907r. This result is a corollary of a more general result on the zeros of the Potts-model partition function ZG(q, {ve}) in the complex antiferromagnetic regime [mid ]1 + ve[mid ] [les ] 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Kotecký–Preiss condition for nonvanishing of a polymer-model partition function. We also show that, for all loopless graphs G of second-largest degree [les ] r, the zeros of PG(q) lie in the disc [mid ]q[mid ] < C(r) + 1. Along the way, I give a simple proof of a generalized (multivariate) Brown–Colbourn conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.

scholarly journals Regions Without Complex Zeros for Chromatic Polynomials on Graphs with Bounded Degree

2008 ◽  
Vol 17 (2) ◽  
pp. 225-238 ◽  
Author(s):  
ROBERTO FERNÁNDEZ ◽  
ALDO PROCACCI
Keyword(s):  
Finite Graph ◽  
Chromatic Polynomial ◽  
Maximal Degree ◽  
Bounded Degree ◽  
Chromatic Polynomials ◽  
Image Position ◽  
Complex Zeros

We prove that the chromatic polynomial$P_\mathbb{G}(q)$of a finite graph$\mathbb{G}$of maximal degree Δ is free of zeros for |q| ≥C*(Δ) withThis improves results by Sokal and Borgs. Furthermore, we present a strengthening of this condition for graphs with no triangle-free vertices.

scholarly journals A Heuristic Method for Certifying Isolated Zeros of Polynomial Systems

2018 ◽  
Author(s):  
Jin-San Cheng ◽  
Xiaojie Dou
Keyword(s):  
Local Minimum ◽  
Heuristic Method ◽  
Polynomial Systems ◽  
Sum Of Squares ◽  
Real System ◽  
Interval Methods ◽  
The Real ◽  
Real Zeros ◽  
Input System ◽  
Complex Zeros

We construct a real square system related to a given over-determined real system. We prove that the simple real zeros of the over-determined system are the simple real zeros of the related square system and the real zeros of the two systems are one-to-one correspondence with the constraint that the value of the sum of squares of the polynomials in the over-determined system at the real zeros is identically zero. After certifying the simple real zeros of the related square system with the interval methods, we assert that the certified zero is a local minimum of the sum of squares of the input polynomials. If the value of the sum of the squares of the input polynomials at the certified zero is equal to zero, it is a zero of the input system. As an application, we also consider the heuristic verification of the isolated zeros of polynomial systems and their multiplicity structures. Notice that a complex system with complex zeros can be transformed into a real system with real zeros.

scholarly journals Chromatic Bounds on Orbital Chromatic Roots

10.37236/4412 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Dae Hyun Kim ◽  
Alexander H. Mun ◽  
Mohamed Omar
Keyword(s):  
Positive Integer ◽  
Real Root ◽  
Chromatic Polynomial ◽  
Outerplanar Graphs ◽  
Chromatic Polynomials ◽  
The Real ◽  
Real Roots ◽  
Chromatic Roots

Given a group $G$ of automorphisms of a graph $\Gamma$, the orbital chromatic polynomial $OP_{\Gamma,G}(x)$ is the polynomial whose value at a positive integer $k$ is the number of orbits of $G$ on proper $k$-colorings of $\Gamma.$ Cameron and Kayibi introduced this polynomial as a means of understanding roots of chromatic polynomials. In this light, they posed a problem asking whether the real roots of the orbital chromatic polynomial of any graph are bounded above by the largest real root of its chromatic polynomial. We resolve this problem in a resounding negative by not only constructing a counterexample, but by providing a process for generating families of counterexamples. We additionally begin the program of finding classes of graphs whose orbital chromatic polynomials have real roots bounded above by the largest real root of their chromatic polynomials; in particular establishing this for many outerplanar graphs.

scholarly journals Planar graphs with maximum degree Δ≥9 are (Δ+1)-edge-choosable—A short proof

2010 ◽  
Vol 310 (21) ◽  
pp. 3049-3051 ◽  
Author(s):  
Nathann Cohen ◽  
Frédéric Havet
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Short Proof

On Certain Functional Identities in EN

1971 ◽  
Vol 23 (2) ◽  
pp. 315-324 ◽  
Author(s):  
A. McD. Mercer
Keyword(s):  
Euclidean Space ◽  
Taylor Series ◽  
Dimensional Space ◽  
The Real ◽  
Higher Dimensional ◽  
Isolated Points ◽  
Functional Identities ◽  
Image Position ◽  
Special Case ◽  
Real Euclidean Space

1. If f is a real-valued function possessing a Taylor series convergent in (a — R, a + R), then it satisfies the following operational identity1.1in which D2 = d2/du2. Furthermore, when g is a solution of y″ + λ2y = 0 in (a – R, a + R), then g is such a function and (1.1) specializes to1.2In this note we generalize these results to the real Euclidean space EN, our conclusions being Theorems 1 and 2 below. Clearly, (1.2) is a special case of (1.1) but in higher-dimensional space it is of interest to allow g, now a solution of1.3to possess singularities at isolated points away from the origin. It is then necessary to consider not only a neighbourhood of the origin but annular regions also.

scholarly journals On a partial theta function and its spectrum

2016 ◽  
Vol 146 (3) ◽  
pp. 609-623 ◽  
Author(s):  
Vladimir Petrov Kostov
Keyword(s):  
Entire Function ◽  
Theta Function ◽  
Local Minimum ◽  
Real Zero ◽  
Positive Real ◽  
The Real ◽  
Real Zeros ◽  
Partial Theta Function

The bivariate series defines a partial theta function. For fixed q (∣q∣ < 1), θ(q, ·) is an entire function. For q ∈ (–1, 0) the function θ(q, ·) has infinitely many negative and infinitely many positive real zeros. There exists a sequence of values of q tending to –1+ such that has a double real zero (the rest of its real zeros being simple). For k odd (respectively, k even) has a local minimum (respectively, maximum) at , and is the rightmost of the real negative zeros of (respectively, for k sufficiently large is the second from the left of the real negative zeros of ). For k sufficiently large one has . One has and .

Location of Zeros of Chromatic and Related Polynomials of Graphs

1994 ◽  
Vol 46 (1) ◽  
pp. 55-80 ◽  
Author(s):  
Francesco Brenti ◽  
Gordon F. Royle ◽  
David G. Wagner
Keyword(s):  
Generating Functions ◽  
Sufficient Conditions ◽  
Combinatorial Identities ◽  
Chromatic Polynomials ◽  
Real Zeros ◽  
Location Of Zeros ◽  
Theoretical Results

AbstractWe consider the location of zeros of four related classes of polynomials, one of which is the class of chromatic polynomials of graphs. All of these polynomials are generating functions of combinatorial interest. Extensive calculations indicate that these polynomials often have only real zeros, and we give a variety of theoretical results which begin to explain this phenomenon. In the course of the investigation we prove a number of interesting combinatorial identities and also give some new sufficient conditions for a polynomial to have only real zeros.

The phase problem in scattering phenomena: the zeros of entire functions and their significance

1978 ◽  
Vol 360 (1700) ◽  
pp. 25-45 ◽  
Keyword(s):  
Entire Functions ◽  
Signal To Noise Ratio ◽  
Phase Problem ◽  
Distribution Of Zeros ◽  
Real Zeros ◽  
Zeros Of Entire Functions ◽  
Functions Of Exponential Type ◽  
Zeros Of Functions ◽  
The Fourier Transform ◽  
Complex Zeros

The paper outlines an approach to the calculation of the phase from intensity data based on the properties of the distribution of zeros of functions of exponential type. This leads to a reinterpretation of such phenomena as Gibbs’ or speckle, which underlines their intrinsic unity. The phase problem is solved for functions which present complex zeros by apodization, i.e. by creating a sufficiently large zero-free area. The method is based on a compromise between signal to noise ratio and resolution and is meaningful provided the apodization required is not too severe. Real zeros, for which the phase problem is trivial, occur only for the special case of eigenfunctions of the Fourier transform

Sign in / Sign up

Export Citation Format

Share Document