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 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.

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 Chromatic Polynomials of Oriented Graphs

10.37236/8240 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Danielle Cox ◽  
Christopher Duffy
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Oriented Graph ◽  
Chromatic Polynomial ◽  
Simple Graph ◽  
Interval Graphs ◽  
Chromatic Polynomials ◽  
Oriented Graphs ◽  
Real Roots ◽  
The Family

The oriented chromatic polynomial of a oriented graph outputs the number of oriented $k$-colourings for any input $k$. We fully classify those oriented graphs for which the oriented graph has the same chromatic polynomial as the underlying simple graph, closing an open problem posed by Sopena. We find that such oriented graphs can be both identified and constructed in polynomial time as they are exactly the family of quasi-transitive oriented co-interval graphs. We study the analytic properties of this polynomial and show that there exist oriented graphs which have chromatic polynomials have roots, including negative real roots,  that cannot be realized as the root of any chromatic polynomial of a simple graph.

Extended Chromatic Polynomials

1972 ◽  
Vol 24 (3) ◽  
pp. 492-501 ◽  
Author(s):  
Andrew Sobczyk ◽  
James O. Gettys
Keyword(s):  
Positive Integer ◽  
Finite Graph ◽  
Chromatic Polynomial ◽  
Chromatic Polynomials ◽  
Vertex Set

Let G be a finite graph with non-empty vertex set (G) and edge set (G) (see [2]). Let λ be a positive integer. Tutte [5] defines a λ-colouring of G as a mapping of (G) into the set Iλ = {1, 2, 3, …, λ} with the property that two ends of any edge are mapped onto distinct integers. The elements of Iλ are commonly called “colours.” If P(G, λ) represents the number of λ-colourings of G, it is well known that P(G, λ) can be expressed as a polynomial in λ. For this reason P(G, λ) is usually referred to as the chromatic polynomial of G.The chromatic polynomial P(G, λ) was first suggested as an approach to the four-colour conjecture. To quote Tutte [5]: ”… many people are specially interested in the value λ = 4.

Pairs of Spherical Mirrors as Prime Focus Correctors for the Anglo-Australian Telescope

1972 ◽  
Vol 2 (3) ◽  
pp. 126-127 ◽  
Author(s):  
N. J. Rumsey
Keyword(s):  
Real Root ◽  
Early Stage ◽  
Spherical Aberration ◽  
Primary Mirror ◽  
The Real ◽  
Real Roots ◽  
Algebraic Analysis ◽  
Cubic Equation ◽  
Spherical Mirrors ◽  
Aberration Corrected

Last year I described pairs of spherical mirrors that remove the coma and astigmatism in the image formed by a paraboloid mirror and leave the spherical aberration corrected. The investigation can be extended to deal with other shapes of primary mirror, for example the hyperboloid primary of the Anglo-Australian Telescope. The algebraic analysis becomes more complicated than for a paraboloid; but it still has the feature that at an early stage a cubic equation has to be solved, each real root of which gives rise to a second cubic. Thus in principle the mathematics could lead to nine solutions. However, it again turns out that not all the roots are real; and even for the real roots not all the solutions are physically useful, because in some cases the final image is virtual, and in others the tertiary mirror lies behind the secondary where light can not reach it. When the primary is a paraboloid, there are three useable solutions all with the property that the field corrector (consisting of the pair of spherical mirrors) can simply be scaled up or down at the user’s pleasure according to the diameter of the field he wishes to photograph. When the primary is of any other shape this is no longer possible.

Improved Iterative Methods for Solving High Order Polynomial Equations

2010 ◽  
Vol 143-144 ◽  
pp. 1122-1126
Author(s):  
Dian Xuan Gong ◽  
Ling Wang ◽  
Chuan An Wei ◽  
Ya Mian Peng
Keyword(s):  
Adaptive Algorithm ◽  
Real Root ◽  
Polynomial Equation ◽  
Polynomial Equations ◽  
The Real ◽  
Real Roots ◽  
Order Polynomial ◽  
Root Isolation ◽  
Sturm Theorem ◽  
Real Root Isolation

Many calculations in engineering and scientific computation can summarized to the problem of solving a polynomial equation. Based on Sturm theorem, an adaptive algorithm for real root isolation is shown. This algorithm will firstly find the isolate interval for all the real roots rapidly. And then approximate the real roots by subdividing the isolate intervals and extracting subintervals each of which contains one real root. This method overcomes all the shortcomings of dichotomy method and iterative method. It doesn’t need to compute derivative values, no need to worry about the initial points, and could find all the real roots out parallelly.

scholarly journals Zero-Free Intervals of Chromatic Polynomials of Mixed Hypergraphs

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 193
Author(s):  
Ruixue Zhang ◽  
Fengming Dong ◽  
Meiqiao Zhang
Keyword(s):  
Positive Integer ◽  
Chromatic Polynomial ◽  
Chromatic Polynomials ◽  
Finite Set ◽  
Graph Function ◽  
Mixed Hypergraph

A mixed hypergraph H is a triple (X,C,D), where X is a finite set and each of C and D is a family of subsets of X. For any positive integer λ, a proper λ-coloring of H is an assignment of λ colors to vertices in H such that each member in C contains at least two vertices assigned the same color and each member in D contains at least two vertices assigned different colors. The chromatic polynomial of H is the graph-function counting the number of distinct proper λ-colorings of H whenever λ is a positive integer. In this article, we show that chromatic polynomials of mixed hypergraphs under certain conditions are zero-free in the intervals (−∞,0) and (0,1), which extends known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs.

scholarly journals Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 169
Author(s):  
Avram Sidi
Keyword(s):  
Nonlinear Equations ◽  
Local Convergence ◽  
Real Root ◽  
Numerical Procedure ◽  
Secant Method ◽  
Convergence Theory ◽  
Simple Complex ◽  
Real Roots ◽  
Complex Roots ◽  
Appl Math

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, 2008), we presented a generalization of the secant method that uses only one evaluation of f(x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence {xn} of approximations to a real root of f(x), where, for n≥k, xn+1=xn−f(xn)/pn,k′(xn), pn,k(x) being the polynomial of degree k that interpolates f(x) at xn,xn−1,…,xn−k, the order sk of this method satisfying 1<sk<2. Clearly, when k=1, this method reduces to the secant method with s1=(1+5)/2. In addition, s1<s2<s3<⋯, such that limk→∞sk=2. In this note, we study the application of this method to simple complex roots of a function f(z). We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.

On Graeffe's Method for Complex Roots of Algebraic Equations

1924 ◽  
Vol 22 (2) ◽  
pp. 83-87 ◽  
Author(s):  
S. Brodetsky ◽  
G. Smeal
Keyword(s):  
Nineteenth Century ◽  
Real Axis ◽  
Practical Method ◽  
Algebraic Equations ◽  
Argand Diagram ◽  
Considerable Difficulty ◽  
The Real ◽  
Real Roots ◽  
Bipolar Coordinates ◽  
Complex Roots

The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.

scholarly journals What Does Productivity Really Mean? Towards an Integrative Paradigm in the Search for Biodiversity-Productivity Relationships

2013 ◽  
Vol 41 (1) ◽  
pp. 36 ◽  
Author(s):  
Liangjun HU ◽  
Qinfeng GUO
Keyword(s):  
Species Richness ◽  
Species Diversity ◽  
Biomass Production ◽  
Real Root ◽  
Spatial Scaling ◽  
Ecological Processes ◽  
The Real ◽  
Community Or ◽  
Per Se ◽  
Underlying Mechanisms

How species diversity relates to productivity remains a major debate. To date, however, the underlying mechanisms that regulate the ecological processes involved are still poorly understood. Three major issues persist in early efforts at resolution. First, in the context that productivity drives species diversity, how the pathways operate is poorly-explained. Second, productivity  per se varies with community or ecosystem maturity. If diversity indeed drives productivity, the criterion of choosing appropriate measures for productivity is not available. Third, spatial scaling suggests that sampling based on small-plots may not be suitable for formulating species richness-productivity relationships (SRPRs). Thus, the long-standing assumption simply linking diversity with productivity and pursuing a generalizing pattern may not be robust. We argue that productivity, though defined as ‘the rate of biomass production’, has been measured in two ways environmental surrogates and biomass production leading to misinterpretations and difficulty in the pursuit of generalizable SRPRs. To tackle these issues, we developed an integrative theoretical paradigm encompassing richer biological and physical contexts and clearly reconciling the major processes of the systems, using proper productivity measures and sampling units. We conclude that loose interpretation and confounding measures of productivity may be the real root of current SRPR inconsistencies and debate.

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