scholarly journals Chromatic Roots of a Ring of Four Cliques

10.37236/638 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
F. M. Dong ◽  
Gordon Royle ◽  
Dave Wagner
Keyword(s):  
Real Number ◽  
Chromatic Polynomial ◽  
Complete Graphs ◽  
Positive Integers ◽  
Chromatic Roots

For any positive integers $a,b,c,d$, let $R_{a,b,c,d}$ be the graph obtained from the complete graphs $K_a, K_b, K_c$ and $K_d$ by adding edges joining every vertex in $K_a$ and $K_c$ to every vertex in $K_b$ and $K_d$. This paper shows that for arbitrary positive integers $a,b,c$ and $d$, every root of the chromatic polynomial of $R_{a,b,c,d}$ is either a real number or a non-real number with its real part equal to $(a+b+c+d-1)/2$.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

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 the distribution (mod 1) of polynomials of a prime variable

1982 ◽  
Vol 85 ◽  
pp. 241-249
Author(s):  
Ming-Chit Liu ◽  
Kai-Man Tsang
Keyword(s):  
Real Number ◽  
Small Positive Number ◽  
Prime Variable ◽  
Positive Integers

Throughout, ε is any small positive number, θ any real number, n, nj, k, N some positive integers and p, pj any primes. By ‖θ‖ we mean the distance from θ to the nearest integer. Write C(ε), C(ε, k) for positive constants which may depend on the quantities indicated inside the parentheses.

Several weighted sum formulas of multiple zeta values

2017 ◽  
Vol 13 (09) ◽  
pp. 2253-2264 ◽  
Author(s):  
Minking Eie ◽  
Wen-Chin Liaw ◽  
Yao Lin Ong
Keyword(s):  
Real Number ◽  
Weighted Sum ◽  
Multiple Zeta Values ◽  
Explicit Evaluation ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Number Formula ◽  
Positive Integers ◽  
Special Case

For a real number [Formula: see text] and positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], we evaluate the sum of multiple zeta values [Formula: see text] explicitly in terms of [Formula: see text] and [Formula: see text]. The special case [Formula: see text] gives an evaluation of [Formula: see text]. An explicit evaluation of the multiple zeta-star value [Formula: see text] is also obtained, as well as some applications to evaluation of multiple zeta values with even arguments.

scholarly journals DENSE SETS OF INTEGERS WITH A PRESCRIBED REPRESENTATION FUNCTION

2011 ◽  
Vol 84 (1) ◽  
pp. 40-43 ◽  
Author(s):  
MIN TANG
Keyword(s):  
Real Number ◽  
Representation Function ◽  
Dense Sets ◽  
Asymptotic Basis ◽  
Positive Integers

AbstractA set A⊆ℤ is called an asymptotic basis of ℤ if all but finitely many integers can be represented as a sum of two elements of A. Let A be an asymptotic basis of integers with prescribed representation function, then how dense A can be? In this paper, we prove that there exist a real number c>0 and an asymptotic basis A with prescribed representation function such that $A(-x,x)\geq c\sqrt {x}$ for infinitely many positive integers x.

scholarly journals On ranges of variants of the divisor functions that are dense

2015 ◽  
Vol 11 (06) ◽  
pp. 1905-1912 ◽  
Author(s):  
Colin Defant
Keyword(s):  
Real Number ◽  
Open Problem ◽  
Dense Subset ◽  
Arithmetic Function ◽  
Divisor Functions ◽  
Positive Integers ◽  
Multiplicative Arithmetic ◽  
Multiplicative Arithmetic Function

For a real number t, let st be the multiplicative arithmetic function defined by [Formula: see text] for all primes p and positive integers α. We show that the range of a function s-r is dense in the interval (0, 1] whenever r ∈ (0, 1]. We then find a constant ηA ≈ 1.9011618 and show that if r > 1, then the range of the function s-r is a dense subset of the interval [Formula: see text] if and only if r ≤ ηA. We end with an open problem.

Probability measures with trivial Stam groups

1982 ◽  
Vol 91 (3) ◽  
pp. 477-484
Author(s):  
Gavin Brown ◽  
William Mohan
Keyword(s):  
Probability Measure ◽  
Real Number ◽  
Real Line ◽  
Probability Measures ◽  
The Real ◽  
Interesting Observation ◽  
Image Position ◽  
Positive Integers

Let μ be a probability measure on the real line ℝ, x a real number and δ(x) the probability atom concentrated at x. Stam made the interesting observation that eitheror else(ii) δ(x)* μn, are mutually singular for all positive integers n.

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 Lacunary ward continuity in 2-normed spaces

Filomat ◽  
2015 ◽  
Vol 29 (10) ◽  
pp. 2257-2263 ◽  
Author(s):  
Huseyin Cakalli ◽  
Sibel Ersan
Keyword(s):  
Real Number ◽  
Normed Space ◽  
Positive Real Number ◽  
Normed Spaces ◽  
Positive Real ◽  
Cauchy Sequences ◽  
Positive Integers

In this paper, we introduce lacunary statistical ward continuity in a 2-normed space. A function f defined on a subset E of a 2-normed space X is lacunary statistically ward continuous if it preserves lacunary statistically quasi-Cauchy sequences of points in E where a sequence (xk) of points in X is lacunary statistically quasi-Cauchy if limr?1 1/hr |{k?Ir : ||xk+1 - xk, z||? ?}| = 0 for every positive real number ? and z ? X, and (kr) is an increasing sequence of positive integers such that k0 = 0 and hr = kr - kr-1 ? ? as r ? ?, Ir = (kr-1, kr]. We investigate not only lacunary statistical ward continuity, but also some other kinds of continuities in 2-normed spaces.

scholarly journals A new two-variable generalization of the chromatic polynomial

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Klaus Dohmen ◽  
André Poenitz ◽  
Peter Tittmann
Keyword(s):  
Chromatic Polynomial ◽  
Complete Graphs ◽  
Matching Polynomial ◽  
Independence Polynomial ◽  
Decomposition Formula ◽  
International Audience ◽  
Complete Bipartite Graphs ◽  
Chromatic Symmetric Function ◽  
Broken Circuit ◽  
Edge Decomposition

International audience We present a two-variable polynomial, which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of a graph. This new polynomial satisfies both an edge decomposition formula and a vertex decomposition formula. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex and one in terms of the lattice of forbidden colorings. We show that the new polynomial may be considered as a specialization of Stanley's chromatic symmetric function. We finally give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that it can be computed in polynomial time for trees and graphs of restricted pathwidth.

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