scholarly journals Perfect dominating sets and perfect domination polynomial of a star graph

2020 ◽  
Vol 8 (4) ◽  
pp. 1751-1755
Author(s):  
Paul Hawkins P. ◽  
Anto A.M. ◽  
Shyla Isac Mary T.
Keyword(s):  
Star Graph ◽  
Dominating Sets ◽  
Domination Polynomial

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.

The Number of 2-dominating Sets, and 2-domination Polynomial of a Graph

2021 ◽  
Vol 42 (4) ◽  
pp. 751-759
Author(s):  
F. Movahedi ◽  
M. H. Akhbari ◽  
S. Alikhani
Keyword(s):  
Dominating Sets ◽  
Domination 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 General Form of Domination Polynomial for Two Types of Graphs Associated to Dihedral Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (2) ◽  
pp. 149-155
Author(s):  
Nabilah Najmuddin ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Complete Graphs ◽  
Dominating Sets ◽  
Dihedral Groups ◽  
Graph Polynomial ◽  
Central Elements ◽  
Domination Polynomial ◽  
Class Graph

A domination polynomial is a type of graph polynomial in which its coefficients represent the number of dominating sets in the graph. There are many researches being done on the domination polynomial of some common types of graphs but not yet for graphs associated to finite groups. Two types of graphs associated to finite groups are the conjugate graph and the conjugacy class graph. A graph of a group G is called a conjugate graph if the vertices are non-central elements of G and two distinct vertices are adjacent if they are conjugate to each other. Meanwhile, a conjugacy class graph of a group G is a graph in which its vertices are the non-central conjugacy classes of G and two distinct vertices are connected if and only if their class cardinalities are not coprime. The conjugate and conjugacy class graph of dihedral groups can be expressed generally as a union of complete graphs on some vertices. In this paper, the domination polynomials are computed for the conjugate and conjugacy class graphs of the dihedral groups.

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 Recurrence Relations and Splitting Formulas for the Domination Polynomial

10.37236/2475 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Tomer Kotek ◽  
James Preen ◽  
Frank Simon ◽  
Peter Tittmann ◽  
Martin Trinks
Keyword(s):  
Generating Function ◽  
Recurrence Relations ◽  
Linear Recurrence ◽  
Dominating Sets ◽  
Special Cases ◽  
Wide Range ◽  
Domination Polynomial ◽  
Arbitrary Graphs

The domination polynomial $D(G,x)$ of a graph $G$ is the generating function of its dominating sets. We prove that $D(G,x)$ satisfies a wide range of reduction formulas. We show linear recurrence relations for $D(G,x)$ for arbitrary graphs and for various special cases. We give splitting formulas for $D(G,x)$ based on articulation vertices, and more generally, on splitting sets of vertices.

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.

The strong nonsplit domination polynomial of some graphs

2020 ◽  
Vol 12 (04) ◽  
pp. 2050053
Author(s):  
D. Kiruba Packiarani ◽  
Y. Therese Sunitha Mary
Keyword(s):  
Domination Number ◽  
Dominating Sets ◽  
Domination Polynomial ◽  
Polynomial Formula

The strong nonsplit domination polynomial of a graph [Formula: see text] of order [Formula: see text] is the polynomial [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]. We obtain some properties of [Formula: see text] and its coefficients. Also, we compute the polynomial for some specific graphs.

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