scholarly journals On Graphs whose Flow Polynomials have Real Roots Only

10.37236/7512 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Fengming Dong
Keyword(s):  
Real Root ◽  
Plane Graph ◽  
Real Roots ◽  
Bridgeless Graph ◽  
Flow Polynomial

Let $G=(V,E)$ be a bridgeless graph. In 2011 Kung and Royle showed that the flow polynomial $F(G,\lambda)$ of $G$ has integral roots only if and only if $G$ is the dual of a chordal and plane graph. In this article, we study whether every graph whose flow polynomial has real roots only is the dual of some chordal and plane graph. We conclude that the answer for this problem is positive if and only if $F(G,\lambda)$ does not have any real root in the interval $(1,2)$. We also prove that for any non-separable and $3$-edge connected $G$, if $G-e$ is also non-separable for each edge $e$ in $G$ and every $3$-edge-cut of $G$ consists of edges incident with some vertex of $G$, then $P(G,\lambda)$ has real roots only if and only if either $G\in \{L,Z_3,K_4\}$ or $F(G,\lambda)$ contains at least $9$ real roots in the interval $(1,2)$, where $L$ is the graph with one vertex and one loop and $Z_3$ is the graph with two vertices and three parallel edges joining these two vertices.

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.

Notes on the Theory of Equations

1939 ◽  
Vol 23 (256) ◽  
pp. 376-379
Author(s):  
E. P. Lewis
Keyword(s):  
Real Root ◽  
Real Roots ◽  
The Third ◽  
Complex Roots ◽  
Image Position

Multiply throughout by a 2 and write y for ax+ b ; the equation becomes where H ≡ ac − b 2, G ≡ a2d − 3abc + 2b3. Since in an equation with real coefficients complex roots occur in conjugate pairs, (i) must have at least one real root; so if α is this root, (i) may be written Accordingly the two remaining roots are also real if But since α satisfies (i), and so Hence if (i) has three real roots, G2 +4H 3 ≤ 0; and clearly, when G2 +4H 3 = 0, two roots are numerically equal to and the third to .

scholarly journals TOPOLOGY OF FAMILIES OF ALGEBRAIC CURVES CONTINUOUSLY DEPENDING ON A PARAMETER, AND APPLICATIONS

2013 ◽  
Vol 23 (07) ◽  
pp. 1591-1610
Author(s):  
JUAN G. ALCAZAR
Keyword(s):  
Real Function ◽  
Real Root ◽  
Algebraic Curves ◽  
Algebraic Surfaces ◽  
Real Interval ◽  
Real Roots ◽  
Level Curves ◽  
The Family ◽  
Critical Sets ◽  
La Rioja

Given a family of algebraic curves whose coefficients depend continuously on a parameter t ∈ U ⊂ ℝ (U a union of real intervals) we address the problem of computing the topology types in the family. Under certain conditions, we provide a method to compute a univariate real function R*(t), with the property that the topology of the family stays invariant along every real interval I ⊂ U not containing any real root of R*(t). So, if R*(t) has finitely many real roots the topology types in the family can be computed. We apply this result to provide bounds on the number of topology types of the family in certain cases, including the important case when the coefficients belong to a Pfaffian chain, and to analyze the existence of certain degeneracies in the real part of a surface whose family of level curves corresponds to the type studied here. The results of the paper can be seen as generalizations of previous works [Alcázar, Applications of level curves to some problems on algebraic surfaces, in Contribuciones Científicas en Honor de Mirian Andrés Gòmez, eds. Lambán, Romero and Rubio (Universidad de La Rioja, Servicio de Publicaciones, 2010), pp. 105–122, Alcázar, Schicho and Sendra, A delineability-based method for computing critical sets of algebraic surfaces, J. Symbolic Comput.42 (2007) 678–691] on the topology of families algebraically depending on a parameter t.

scholarly journals On Graphs Having no Flow Roots in the Interval $(1,2)$

10.37236/3841 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
F.M. Dong
Keyword(s):  
Bridgeless Graph ◽  
Flow Polynomial

For any graph $G$, let $W(G)$ be the set of vertices in $G$ of degrees larger than 3. We show that for any bridgeless graph $G$, if $W(G)$ is dominated by some component of $G - W(G)$, then $F(G,\lambda)$ has no roots in the interval (1,2), where $F(G,\lambda)$ is the flow polynomial of $G$. This result generalizes the known result that $F(G,\lambda)$ has no roots in (1,2) whenever $|W(G)| \leq 2$. We also give some constructions to generate graphs whose flow polynomials have no roots in $(1,2)$.

scholarly journals Real Root Polynomials and Real Root Preserving Transformations

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Suchada Pongprasert ◽  
Kanyarat Chaengsisai ◽  
Wuttichai Kaewleamthong ◽  
Puttarawadee Sriphrom
Keyword(s):  
Real World ◽  
Real Root ◽  
Analogous Result ◽  
Complex Root ◽  
Linear Transformations ◽  
Real Numbers ◽  
Real Roots ◽  
Fundamental Theorem Of Algebra ◽  
Roots Of Polynomials ◽  
Complex Coefficients

Polynomials can be used to represent real-world situations, and their roots have real-world meanings when they are real numbers. The fundamental theorem of algebra tells us that every nonconstant polynomial p with complex coefficients has a complex root. However, no analogous result holds for guaranteeing that a real root exists to p if we restrict the coefficients to be real. Let n ≥ 1 and P n be the vector space of all polynomials of degree n or less with real coefficients. In this article, we give explicit forms of polynomials in P n such that all of their roots are real. Furthermore, we present explicit forms of linear transformations on P n which preserve real roots of polynomials in a certain subset of P n .

scholarly journals V. A New Method of resolving Cubic Equations

1805 ◽  
Vol 5 (1) ◽  
pp. 99-116 ◽  
Author(s):  
James Ivory
Keyword(s):  
Real Root ◽  
The Other ◽  
New Method ◽  
Real Roots ◽  
The One ◽  
Image Position

1. I Divide cubic equations into two varieties or species: the one, comprehending all cubic equations with three real roots; the other, all those with only one real root.2. Let φ denote any angle whatever, and let τ = tan φ, the radius being unity: let also : then from the doctrine of angular sections we havewhich being reduced to the form of an equation, isZ3-3τZ2-3Z+τ=0.

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.

A Comparative Study of Two Real Root Isolation Methods

2005 ◽  
Vol 10 (4) ◽  
pp. 297-304 ◽  
Author(s):  
A. G. Akritas ◽  
A. W. Strzebonski
Keyword(s):  
Continued Fractions ◽  
Recent Progress ◽  
Real Root ◽  
Bisection Method ◽  
Memory Usage ◽  
Real Roots ◽  
Isolation Methods ◽  
Root Isolation ◽  
High Degree ◽  
Real Root Isolation

Recent progress in polynomial elimination has rendered the computation of the real roots of ill-conditioned polynomials of high degree (over 1000) with huge coefficients (several thousand digits) a critical operation in computer algebra. To rise to the occasion, the only method-candidate that has been considered by various authors for modification and improvement has been the Collins-Akritas bisection method [1], which is a based on a variation of Vincent’s theorem [2]. The most recent example is the paper by Rouillier and Zimmermann [3], where the authors present “... a new algorithm, which is optimal in terms of memory usage and as fast as both Collins and Akritas’ algorithm and Krandick variant ...” [3] In this paper we compare our own continued fractions method CF [4] (which is directly based on Vincent’s theorem) with the best bisection method REL described in [3]. Experimentation with the data presented in [3] showed that, with respect to time, our continued fractions method CF is by far superior to REL, whereas the two are about equal with respect to space.

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 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.

Sign in / Sign up

Export Citation Format

Share Document