Some families of graphs with no nonzero real domination roots

Let G be a simple graph of order n. The domination polynomial is the generating polynomial for the number of dominating sets of G of each cardinality. A root of this polynomial is called a domination root of G. Obviously 0 is a domination root of every graph G. In the study of the domination roots of graphs, this naturally raises the question: which graphs have no nonzero real domination roots? In this paper we present some families of graphs whose have this property.Comment: 13 pages, 5 figures. arXiv admin note: text overlap with arXiv:1401.209

On the Domination Polynomials of Friendship Graphs

Let $G$ be a simple graph of order $n$. The {\em domination polynomial} of $G$ 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 $i$. Let $n$ be any positive integer and $F_n$ be the Friendship graph with $2n + 1$ vertices and $3n$ edges, formed by the join of $K_{1}$ with $nK_{2}$. 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 $n\geq 2$, $F_n$ is not $\mathcal{D}$-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 $B_n$, formed by joining $n$ copies of the cycle graph $C_4$ with a common edge.Comment: 16 pages, 7 figures. New version of paper entitled "On $\mathcal{D}$-equivalence class of friendship graphs

Domination polynomials of k-tree related graphs

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ 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 study the domination polynomials of several classes of $k$-tree related graphs. Also, we present families of these kind of graphs, whose domination polynomial have no nonzero real roots

An atlas of domination polynomials of graphs of order at most six

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 $G$. The roots of domination polynomial is called domination roots. In this article, we compute the domination polynomial and domination roots of all graphs of order less than or equal to 6, and show them in the tables.Comment: This atlas has published in the PhD thesis of the first author in 2009 and also in book "Dominating sets and domination polynomials of graphs: Domination polynomial: A new graph polynomial", LAMBERT Academic Publishing, ISBN: 9783847344827 (2012). Thanks to ArXiv for letting us to publish it here for more acces

The number of dominating kk-sets of paths, cycles and wheels

We give a shorter proof of the recurrence relation for the domination polynomial $\gamma (P_{n},t)$ and for the number $\gamma _{k}(P_{n})$ of dominating $k$-sets of the path with $n$ vertices. For every positive integers $n$ and $k,$ numbers $\gamma _{k}(P_{n})$ are determined solving a problem posed by S. Alikhani in CID 2015. Moreover, the numbers of dominating $k$-sets $\gamma _{k}(C_{n})$ of cycles and $\gamma _{k}(W_{n})$ of wheels with $n$ vertices are computed.Comment: 13 page

Total domination polynomials of graphs

Given a graph $G$, a total dominating set $D_t$ is a vertex set that every vertex of $G$ is adjacent to some vertices of $D_t$ and let $d_t(G,i)$ be the number of all total dominating sets with size $i$. The total domination polynomial, defined as $D_t(G,x)=\sum\limits_{i=1}^{| V(G)|} d_t(G,i)x^i$, recently has been one of the considerable extended research in the field of domination theory. In this paper, we obtain the vertex-reduction and edge-reduction formulas of total domination polynomials. As consequences, we give the total domination polynomials for paths and cycles. Additionally, we determine the sharp upper bounds of total domination polynomials for trees and characterize the corresponding graphs attaining such bounds. Finally, we use the reduction-formulas to investigate the relations between vertex sets and total domination polynomials in $G$

On D\mathcal{D}-equivalence classes of some graphs

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=1}^n d(G,i) x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. The $n$-barbell graph $Bar_n$ with $2n$ vertices, is formed by joining two copies of a complete graph $K_n$ by a single edge. We prove that for every $n\geq 2$, $Bar_n$ is not $\mathcal{D}$-unique, that is, there is another non-isomorphic graph with the same domination polynomial. More precisely, we show that for every $n$, the $\mathcal{D}$-equivalence class of barbell graph, $[Bar_n]$, contains many graphs, which one of them is the complement of book graph of order $n-1$, $B_{n-1}^c$. Also we present many families of graphs in $\mathcal{D}$-equivalence class of $K_{n_1}\cup K_{n_2}\cup \cdots\cup K_{n_k}$.Comment: 9 pages, 5 figure

Some new results on the total domination polynomial of a graph

Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of $G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and is denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)x^i$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. A root of $D_t(G,x)$ is called a total domination root of $G$. An irrelevant edge of $D_t(G,x)$ is an edge $e \in E$, such that $D_t(G, x) = D_t(G\setminus e, x)$. In this paper, we characterize edges possessing this property. Also we obtain some results for the number of total dominating sets of a regular graph. Finally, we study graphs with exactly two total domination roots $\{-3,0\}$, $\{-2,0\}$ and $\{-1,0\}$.Comment: 12 pages, 7 figure

Final title: "More on domination polynomial and domination root" Previous title: "Graphs with domination roots in the right half-plane"

Let $G$ be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x) =\sum d(G, i)x^i, where d(G,i) is the number of dominating sets of G of size i. Every root of D(G,x) is called the domination root of G. It is clear that (0,\infty) is zero free interval for domination polynomial of a graph. It is interesting to investigate graphs which have complex domination roots with positive real parts. In this paper, we first investigate complexity of the domination polynomial at specific points. Then we present and investigate some families of graphs whose complex domination roots have positive real part.Comment: 18 Pages, 6 Figures. To appear in Ars Combi

Recurrence relations and splitting formulas for the domination polynomial

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