Some new results on domination roots of a graph

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G,\lambda)=\sum_{i=0}^{n} d(G,i) \lambda^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Every root of $D(G,\lambda)$ is called the domination root of $G$. We present families of graphs whose their domination polynomial have no nonzero real roots. We observe that these graphs have complex domination roots with positive real part. Then, we consider the lexicographic product of two graphs and obtain a formula for domination polynomial of this product. Using this product, we construct a family of graphs which their domination roots are dense in all of $\mathbb{C}$

A study of reciprocal Dunford-Pettis-like properties on Banach spaces

In this article, we study the relationship between $p$-$(V)$ subsets and p-$V^*$ subsets of dual spaces. We investigate the Banach space X with the property that adjoint every $p$-convergent operator $T: X \rightarrow Y$ is weakly $q$-compact, for every Banach space $Y$. Moreover, we define the notion of $q$-reciprocal Dunford-Pettis$\^*$property of order $p$ on Banach spaces and obtain a characterization of Banach spaces with this property. The stability of reciprocal Dunford-Pettis property of order $p$ for the projective tensor product is given

On pseudo weakly compact operators of order P P

In this paper, we introduce the concept of a pseudo weakly compact operator of order $p$ between Banach spaces. Also we study the notion of $p$-Dunford-Pettis relatively compact property which is in "general" weaker than the Dunford-Pettis relatively compact property and gives some characterizations of Banach spaces which have this property. Moreover, by using the notion of $p$-Right subsets of a dual Banach space, we study the concepts of $p$-sequentially Right and weak $p$-sequentially Right properties on Banach spaces. Furthermore, we obtain some suitable conditions on Banach spaces $X$ and $Y$ such that projective tensor and injective tensor products between $X$ and $Y$ have the $p$-sequentially Right property.\ Finally, we introduce two properties for the Banach spaces, namely $p$-sequentially Right$^{\ast}$ and weak $p$-sequentially Right$^{\ast}$ properties and obtain some characterizations of these properties

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

Domination polynomial of clique cover product of 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 $i$. For two graphs $G$ and $H$, let $\mathcal{C} = \{C_1,C_2, \cdots, C_k\}$ be a clique cover of $G$ and $U\subseteq V(H)$. We consider clique cover product which denoted by $G^\mathcal{C} \star H^U$ and obtained from $G$ as follows: for each clique $C_i \in \mathcal{C}$, add a copy of the graph $H$ and join every vertex of $C_i$ to every vertex of $U$. We prove that the domination polynomial of clique cover product $G^\mathcal{C} \star H^{V(H)}$ or simply $G^\mathcal{C} \star H$ is $$D(G^\mathcal{C} \star H,x)=\prod_{i=1}^k\Big [\big((1+x)^{n_i}-1\big)(1+x)^{|V(H)|}+D(H,x)\Big],$$ where each clique $C_i \in \mathcal{C}$ has $n_i$ vertices. As results, we study the $\mathcal{D}$-equivalence classes of some families of graphs. Also we completely describe the $\mathcal{D}$-equivalence classes of friendship graphs constructed by coalescence $n$ copies of the cycle graph of length three with a common vertex.Comment: 11 pages, 5 figure

The distinguishing index of graphs with at least one cycle is not more than its distinguishing number

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex (edge) labeling with $d$ labels that is preserved only by the trivial automorphism. It is known that for every graph $G$ we have $D'(G) \leq D(G) + 1$. The complete characterization of finite trees $T$ with $D'(T)=D(T)+ 1$ has been given recently. In this note we show that if $G$ is a finite connected graph with at least one cycle, then $D'(G)\leq D(G)$. Finally, we characterize all connected graphs for which $D'(G) \leq D(G)$

The distinguishing number and the distinguishing index of co-normal product of two graphs

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. The co-normal product $G\star H$ of two graphs $G$ and $H$ is the graph with vertex set $V (G)\times V (H)$ and edge set $\{\{(x_1, x_2), (y_1, y_2)\} | x_1y_1 \in E(G) ~{\rm or}~x_2y_2 \in E(H)\}$. In this paper we study the distinguishing number and the distinguishing index of the co-normal product of two graphs. We prove that for every $k \geq 3$, the $k$-th co-normal power of a connected graph $G$ with no false twin vertex and no dominating vertex, has the distinguishing number and the distinguishing index equal two.Comment: 8 pages. arXiv admin note: text overlap with arXiv:1703.0187

Distinguishing number and distinguishing index of join of two graphs

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. In this paper we study the distinguishing number and the distinguishing index of join of two graphs $G$ and $H$, i.e., $G+H$. We prove that $0\leq D(G+H)-max\{D(G),D(H)\}\leq z$, where $z$ is depends of the number of some induced subgraphs generated by some suitable partitions of $V(G)$ and $V(H)$. Also, we prove that if $G$ is a connected graph of order $n \geq 2$, then $D'(G+ \cdots +G)=2$, except $D'(K_2+K_2)=3$.Comment: 12 pages, 2 figure

Total dominator chromatic number of some operations on a graph

Let $G$ be a simple graph. A total dominator coloring of $G$ is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $\chi_d^t(G)$ of $G$ is the minimum number of colors among all total dominator coloring of $G$. In this paper, we examine the effects on $\chi_d^t(G)$ when $G$ is modified by operations on vertex and edge of $G$.Comment: 10 pages, 5 figures. arXiv admin note: text overlap with arXiv:1511.0165

The distinguishing number (index) and the domination number of a graph

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. A set $S$ of vertices in $G$ is a dominating set of $G$ if every vertex of $V(G)\setminus S$ is adjacent to some vertex in $S$. The minimum cardinality of a dominating set of $G$ is the domination number of $G$ and denoted by $\gamma (G)$. In this paper, we obtain some upper bounds for the distinguishing number and the distinguishing index of a graph based on its domination number.Comment: 8 pages, 2 figure