Domination Polynomial Research Articles

The Domination Polynomial of a Graph at −1

Let G be a simple graph. The domination polynomial of a graph G of order n is the polynomial $${D(G,x)=\sum_{i=0}^{n} d(G,i) x^{i}}$$ , where d(G, i) is the number of dominating sets of G of size i. In this article we investigate the domination polynomial at ?1. We give a construction showing that for each odd number n there is a connected graph G with D(G, ?1) = n.

On the values of independence and domination polynomials at specific points

Let $G$ be a simple graph of order $n$‎. ‎We consider the‎ ‎independence polynomial and the domination polynomial of a graph‎ ‎$G$‎. ‎The value of a graph polynomial at a specific point can give‎ ‎sometimes a very surprising information about the structure of the‎ ‎graph‎. ‎In this paper we investigate independence and domination‎ ‎polynomial at $-1$ and $1$‎.

Algebraic Integers as Chromatic and Domination Roots

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.

Domination Reliability

In this paper, we propose a new network reliability measure for some particular kind of service networks, which we refer to as domination reliability. We relate this new reliability measure to the domination polynomial of a graph and the coverage probability of a hypergraph. We derive explicit and recursive formulæ for domination reliability and its associated domination reliability polynomial, deduce an analogue of Whitney's broken circuit theorem, and prove that computing domination reliability is NP-hard.

Dominating Sets and Domination Polynomials of Square Of Cycles

Dominating Sets and Domination Polynomials of Square Of Cycles

On the graphs with four distinct domination roots

The domination polynomial of a graph G of order n is the polynomial , where d(G, i) is the number of dominating vertex sets of G with cardinality i. A root of D(G, x) is called a domination root of G. In this paper, we characterize graphs with exactly four distinct domination roots .

Domination polynomials of cubic graphs of order 10

Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)=\sumi=g(G)n d(G,i) xi, where d(G,i) is the number of dominating sets of G of size i, and g(G) is the domination number of G. In this paper we study the domination polynomials of cubic graphs of order 10. As a consequence, we show that the Petersen graph is determined uniquely by its domination polynomial.

Comparisons and bounds for expected lifetimes of reliability systems

Sharp bounds on expectations of lifetimes of coherent and mixed systems composed of elements with independent and either identically or non-identically distributed lifetimes are expressed in terms of expected lifetimes of components. Similar evaluations are concluded for the respective mean residual lifetimes. In the IID case, improved inequalities dependent on a concentration parameter connected to the Gini dispersion index are obtained. The results can be used to compare systems with component lifetimes ordered in the convex ordering. In the INID case, some refined bounds are derived in terms of the expected lifetimes of series systems of smaller sizes, and the expected lifetime of single unit for the equivalent systems with IID components. The latter can be further simplified in the case of weak Schur-concavity and Schur-convexity of the system generalized domination polynomial.

Characterization of graphs using domination polynomials

Let G be a simple graph of order n . The domination polynomial of G is the polynomial D ( G , x ) = ∑ i = 1 n d ( G , i ) x i , where d ( G , i ) is the number of dominating sets of G of size i . A root of D ( G , x ) is called a domination root of G . We denote the set of distinct domination roots by Z ( D ( G , x ) ) . Two graphs G and H are said to be D -equivalent, written as G ∼ H , if D ( G , x ) = D ( H , x ) . The D -equivalence class of G is [ G ] = { H : H ∼ G } . A graph G is said to be D -unique if [ G ] = { G } . In this paper, we show that if a graph G has two distinct domination roots, then Z ( D ( G , x ) ) = { − 2 , 0 } . Also, if G is a graph with no pendant vertex and has three distinct domination roots, then Z ( D ( G , x ) ) ⊆ { 0 , − 2 ± 2 i , − 3 ± 3 i 2 } . Also, we study the D -equivalence classes of some certain graphs. It is shown that if n ≡ 0 , 2 ( mod 3 ) , then C n is D -unique, and if n ≡ 0 ( mod 3 ) , then [ P n ] consists of exactly two graphs.

Dominatind sets and domination polynomials of certain graphs. II

The domination polynomial of a graph \(G\) of order \(n\) is the polynomial \(D(G,x) = \sum _{i=\gamma(G)}^n d(G,i)x^i\), where \(d(G,i)\) is the number of dominating sets of \(G\) of size \(i\), and \(\gamma (G)\) is the domination number of \(G\). In this paper, we obtain some properties of the coefficients of \(D(G,x)\). Also, by study of the dominating sets and the domination polynomials of specific graphs denoted by \(G^{\prime}(m)\), we obtain a relationship between the domination polynomial of graphs containing an induced path of length at least three, and the domination polynomial of related graphs obtained by replacing the path by shorter path. As examples of graphs \(G^{\prime}(m)\), we study the dominating sets and domination polynomials of cycles and generalized theta graphs. Finally, we show that, if \(n \equiv 0,2(mod\, 3)\) and \(D(G,x) = D(C_n, x)\), then \(G = C_n\).

Dominating Sets and Domination Polynomials of Paths

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.