scholarly journals Algebraic Integers as Chromatic and Domination Roots

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Saeid Alikhani ◽  
Roslan Hasni
Keyword(s):  
Monic Polynomial ◽  
Chromatic Polynomial ◽  
Simple Graph ◽  
Dominating Sets ◽  
Algebraic Integers ◽  
Integer Coefficients ◽  
Chromatic Root ◽  
Domination Polynomial

Let G be a simple graph of order n and λ∈ℕ. A mapping f:V(G)→{1,2,…,λ} is called a λ-colouring of G if f(u)≠f(v) whenever the vertices u and v are adjacent in G. The number of distinct λ-colourings of G, denoted by P(G,λ), is called the chromatic polynomial of G. The domination polynomial of G is the polynomial D(G,λ)=∑i=1nd(G,i)λi, where d(G,i) is the number of dominating sets of G of size i. Every root of P(G,λ) and D(G,λ) is called the chromatic root and the domination root of G, respectively. Since chromatic polynomial and domination polynomial are monic polynomial with integer coefficients, its zeros are algebraic integers. This naturally raises the question: which algebraic integers can occur as zeros of chromatic and domination polynomials? In this paper, we state some properties of this kind of algebraic integers.

Strong nonsplit dominating sets and strong nonsplit domination polynomial of complement of paths

2020 ◽  
Vol 12 (06) ◽  
pp. 2050082
Author(s):  
D. Kiruba Packiarani ◽  
Y. Therese Sunitha Mary
Keyword(s):  
Domination Number ◽  
Recursive Formula ◽  
Simple Graph ◽  
Dominating Sets ◽  
The Family ◽  
Domination Polynomial

Let [Formula: see text] be a simple graph of order [Formula: see text]. The strong nonsplit domination polynomial of a graph [Formula: see text] is [Formula: see text] where [Formula: see text] is the number of strong nonsplit dominating sets of [Formula: see text] of size [Formula: see text] and [Formula: see text] is the strong nonsplit domination number of [Formula: see text]. Let [Formula: see text] be the family of strong nonsplit dominating sets of a complement of a path [Formula: see text] ([Formula: see text]) with cardinality [Formula: see text], and let [Formula: see text]. In this paper, we construct [Formula: see text], the recursive formula for [Formula: see text] and [Formula: see text], the strong nonsplit domination polynomial of [Formula: see text]. Also, we obtain some properties of the coefficients of this polynomial.

scholarly journals ON THE ROOTS OF TOTAL DOMINATION POLYNOMIAL OF GRAPHS, II

2019 ◽  
pp. 659
Author(s):  
Saeid Alikhani ◽  
Nasrin Jafari
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Domination Polynomial

Let $G = (V, E)$ be a simple graph of order $n$. A  total dominating set of $G$ is a subset $D$ of $V$, such that every vertex of $V$ is adjacent to at least one vertex in  $D$. The total domination number of $G$ is  minimum cardinality of  total dominating set in $G$ and is denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. In this paper, we study roots of the total domination polynomial of some graphs.  We show that  all roots of $D_t(G, x)$ lie in the circle with center $(-1, 0)$ and radius $\sqrt[\delta]{2^n-1}$, where $\delta$ is the minimum degree of $G$. As a consequence, we prove that if $\delta\geq \frac{2n}{3}$,  then every integer root of $D_t(G, x)$ lies in the set $\{-3,-2,-1,0\}$.

scholarly journals Connected Domination Polynomial of Graphs

2018 ◽  
Vol 60 (1) ◽  
pp. 103-121 ◽  
Author(s):  
D. A. Mojdeh ◽  
A. S. Emadi
Keyword(s):  
Domination Number ◽  
Simple Graph ◽  
Dominating Sets ◽  
Connected Dominating Sets ◽  
Connected Domination Number ◽  
Domination Polynomial

Abstract Let G be a simple graph of order n. The connected domination polynomial of G is the polynomial $D_c \left( {G,x} \right) = \sum\nolimits_{i = \gamma _c \left( G \right)}^{\left| {V\left( G \right)} \right|} {d_c \left( {G,i} \right)x^i }$ , where dc(G,i) is the number of connected dominating sets of G of size i and γc(G) is the connected domination number of G. In this paper we study Dc(G,x) of any graph. We classify many families of graphs by studying their connected domination polynomial.

scholarly journals On the Domination Polynomial of Some Graph Operations

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Saeid Alikhani
Keyword(s):  
Simple Graph ◽  
Dominating Sets ◽  
Graph Operations ◽  
Domination Polynomial

Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,λ)=∑i=0n‍d(G,i)λi, where d(G,i) is the number of dominating sets of G of size i. Every root of D(G,λ) is called the domination root of G. In this paper, we study the domination polynomial of some graph operations.

scholarly journals Dominating Sets and Domination Polynomials of Paths

2009 ◽  
Vol 2009 ◽  
pp. 1-10 ◽  
Author(s):  
Saeid Alikhani ◽  
Yee-Hock Peng
Keyword(s):  
Dominating Set ◽  
Recursive Formula ◽  
Simple Graph ◽  
Dominating Sets ◽  
The Family ◽  
Domination Polynomial

LetG=(V,E)be a simple graph. A setS⊆Vis a dominating set ofG, if every vertex inV\Sis adjacent to at least one vertex inS. Let𝒫nibe the family of all dominating sets of a pathPnwith cardinalityi, and letd(Pn,j)=|𝒫nj|. In this paper, we construct𝒫ni, and obtain a recursive formula ford(Pn,i). Using this recursive formula, we consider the polynomialD(Pn,x)=∑i=⌈n/3⌉nd(Pn,i)xi, which we call domination polynomial of paths and obtain some properties of this polynomial.

scholarly journals Domination Polynomials of k-Tree Related Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Somayeh Jahari ◽  
Saeid Alikhani
Keyword(s):  
Domination Number ◽  
Simple Graph ◽  
Dominating Sets ◽  
Real Roots ◽  
Domination Polynomial

Let G be a simple graph of order n. The domination polynomial of G is the polynomial DG,x=∑i=γ(G)nd(G,i)xi, where d(G, i) is the number of dominating sets of G of size i and γ(G) is the domination number of G. In this paper, we study the domination polynomials of several classes of k-tree related graphs. Also, we present families of these kinds of graphs, whose domination polynomials have no nonzero real roots.

scholarly journals On the domination polynomials of friendship graphs

Filomat ◽  
2016 ◽  
Vol 30 (1) ◽  
pp. 169-178 ◽  
Author(s):  
Saeid Alikhani ◽  
Jason Brown ◽  
Somayeh Jahari
Keyword(s):  
Positive Integer ◽  
Simple Graph ◽  
Dominating Sets ◽  
Common Edge ◽  
Related Family ◽  
Isomorphic Graph ◽  
The Family ◽  
Domination Polynomial ◽  
Cycle Graph

Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)= n?i=0 d(G,i)xi, where d(G,i) is the number of dominating sets of G of size i. Let n be any positive integer and Fn be the Friendship graph with 2n + 1 vertices and 3n edges, formed by the join of K1 with nK2. We study the domination polynomials of this family of graphs, and in particular examine the domination roots of the family, and find the limiting curve for the roots. We also show that for every n > 2, Fn is not D-unique, that is, there is another non-isomorphic graph with the same domination polynomial. Also we construct some families of graphs whose real domination roots are only -2 and 0. Finally, we conclude by discussing the domination polynomials of a related family of graphs, the n-book graphs Bn, formed by joining n copies of the cycle graph C4 with a common edge.

scholarly journals Regular polygons, Morgan-Voyce polynomials, and Chebyshev polynomials

2021 ◽  
Vol 27 (2) ◽  
pp. 79-87
Author(s):  
Jorma K. Merikoski ◽  
Keyword(s):  
Chebyshev Polynomials ◽  
Monic Polynomial ◽  
Regular Polygons ◽  
Integer Coefficients

We say that a monic polynomial with integer coefficients is a polygomial if its each zero is obtained by squaring the edge or a diagonal of a regular n-gon with unit circumradius. We find connections of certain polygomials with Morgan-Voyce polynomials and further with Chebyshev polynomials of second kind.

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.

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