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.

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 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 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.

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.

Bounds on the k-tuple domatic number of a graph

2011 ◽  
Vol 61 (6) ◽  
Author(s):  
Lutz Volkmann
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Dominating Set ◽  
Simple Graph ◽  
Dominating Sets ◽  
Domatic Number ◽  
Vertex Set

AbstractLet k be a positive integer, and let G be a simple graph with vertex set V (G). A vertex of a graph G dominates itself and all vertices adjacent to it. A subset S ⊆ V (G) is a k-tuple dominating set of G if each vertex of V (G) is dominated by at least k vertices in S. The k-tuple domatic number of G is the largest number of sets in a partition of V (G) into k-tuple dominating sets.In this paper, we present a lower bound on the k-tuple domatic number, and we establish Nordhaus-Gaddum inequalities. Some of our results extends those for the classical domatic number.

scholarly journals Construction of Dominating Sets of Certain Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Saeid Alikhani ◽  
Yee-Hock Peng
Keyword(s):  
Dominating Set ◽  
Recursive Formula ◽  
Simple Graph ◽  
Dominating Sets ◽  
The Family

Let G be a simple graph. A set S⊆V is a dominating set of G, if every vertex in V∖S is adjacent to at least one vertex in S. We denote the family of dominating sets of a graph G with cardinality i by 𝒟(G,i). In this paper we introduce graphs with specific constructions, which are denoted by G(m). We construct the dominating sets of G(m) by dominating sets of graphs G(m−1), G(m−2), and G(m−3). As an example of G(m), we consider 𝒟(Pn,i). As a consequence, we obtain the recursive formula for the number of dominating sets of G(m).

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.

Family Justice, “Asian Values” and the Politics of Ambivalence in Singapore

2018 ◽  
Vol 46 (4-5) ◽  
pp. 467-483
Author(s):  
Daniel P.S. Goh
Keyword(s):  
Nuclear Family ◽  
Family Values ◽  
Policy Changes ◽  
Asian Values ◽  
Related Family ◽  
New Family ◽  
Rights Of The Child ◽  
The Family ◽  
Family Types ◽  
Discursive Field

Abstract In recent years, Singapore made significant reforms towards the establishment of a dedicated family justice system, setting up the Family Justice Courts and enacting new laws to better manage the divorce process and the protection of children. Related policy changes have also been implemented to provide and support families that were previously considered non-traditional and even deviant. Rhetorically, the state, led by the long-ruling People’s Action Party, continues to champion the modern nuclear family with heterosexual marriage at its core as the normal “traditional” form of the family and the bedrock of conservative “Asian values” defining society and politics in Singapore. However, what the judiciary espouse as the new family justice paradigm and the related family justice practices, together with the shifts in social policy towards different family types, are changing the texture of the dominant conservatism rallied by “Asian values” discourse. This article locates and analyses the incipient paradigm shift in the rising pluralism of family forms and the influence of international legal developments in protecting the rights of the child and interventionist family law. By attempting to bridge the Weberian chasm of doing sociology as a vocation and doing politics as a vocation (as an opposition Member of Parliament), I show that the family justice paradigm has opened up the discursive field on the family and produce the politics of ambivalence caught between family justice and Asian family values. I argue for a relational family justice paradigm as a way to move beyond the politics of ambivalence.

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