Let G = (V, E) be a simple graph. The DOMINATION POLYNOMIAL of G is the POLYNOMIAL D (G, x)= Sn i=0 d (G, i) xi, where d(G, i) is the number of dominating sets of G of size i. In this paper, we present some new approaches for computation of DOMINATION POLYNOMIAL of specific graphs.

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 some vertices of $D$. The total DOMINATION number of $G$ is equal to the minimum cardinality of a 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 $i$. Two graphs $G$ and $H$ are said to be total dominating equivalent or simply $\mathcal{D}_t$-equivalent, if $D_t(G,x)=D_t(H,x)$. The equivalence class of $G$, denoted $[G]$, is the set of all graphs $\mathcal{D}_t$-equivalent to $G$. A POLYNOMIAL $\sum_{k=0}^n a_kx^k$ is called unimodal if the sequence of its coefficients is unimodal, that means there is some $k \in \{0, 1, \ldots , n\}$, such that $a_0 \leq \ldots \leq a_{k-1} \leq a_k\geq a_{k+1} \geq \ldots \geq a_n$. In this paper, we investigate $\mathcal{D}_t$-equivalence classes of some graphs. Also, we introduce some families of graphs whose total DOMINATION POLYNOMIALs are unimodal. The $\mathcal{D}_t$-equivalence classes of graphs of order $\leq 6$ are presented in the appendix.

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 some vertices of $D$. The total DOMINATION number of $G$ is equal to the minimum cardinality of a 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 $i$. Two graphs $G$ and $H$ are said to be total dominating equivalent or simply $\mathcal{D}_t$-equivalent, if $D_t(G,x)=D_t(H,x)$. The equivalence class of $G$, denoted $[G]$, is the set of all graphs $\mathcal{D}_t$-equivalent to $G$. A POLYNOMIAL $\sum_{k=0}^n a_kx^k$ is called unimodal if the sequence of its coefficients is unimodal, that means there is some $k \in \{0, 1, \ldots , n\}$, such that $a_0 \leq \ldots \leq a_{k-1} \leq a_k\geq a_{k+1} \geq \ldots \geq a_n$. In this paper, we investigate $\mathcal{D}_t$-equivalence classes of some graphs. Also, we introduce some families of graphs whose total DOMINATION POLYNOMIALs are unimodal. The $\mathcal{D}_t$-equivalence classes of graphs of order $\leq 6$ are presented in the appendix.

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.

Let G be a simple graph of order n. The DOMINATION polyno-mial of G is the POLYNOMIAL D (G; x) =åni=g(G) 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 present some families of graphs whose DOMINATION POLYNOMIALs are unimodal.

‎A dominating set of a graph $G$ is a subset $D$ of vertices such that every vertex outside $D$ has a neighbor in $D$‎. ‎The DOMINATION number of $G$‎, ‎denoted by $\gamma(G)$‎, ‎is the minimum cardinality amongst all dominating sets of $G$‎. ‎The DOMINATION entropy of $G$‎, ‎denoted by $I_{dom}(G)$ is defined as $I_{dom}(G)=-\sum_{i=1}^k\frac{d_i(G)}{\gamma_S(G)}\log (\frac{d_i(G)}{\gamma_S(G)})$‎, ‎where $\gamma_S(G)$ is the number of all dominating sets of $G$ and $d_i(G)$ is the number of dominating sets of cardinality $i$‎. ‎A graph $G$ is $C_4$-free if it does not contain a $4$-cycle as a subgraph‎. ‎In this note we first determine the DOMINATION entropy in the graphs whose complements are $C_4$-free‎. ‎We then propose an algorithm that computes the DOMINATION entropy in any given graph‎. ‎We also consider circulant graphs $G$ and determine $d_i(G)$ under certain conditions on $i$‎.

A graph G is called P4-free, if G does not contain an induced subgraph P4. The DOMINATION POLYNOMIAL of a graph G of order n is the POLYNOMIAL D (G, x)=Sni=1d (G, i) xi, where d (G, i) is the number of dominating sets of G of size i. Every root of D (G, x) is called a DOMINATION root of G. In this paper we state and prove formula for the DOMINATION POLYNOMIAL of non P4-free graphs. Also, we pose a conjecture about DOMINATION roots of these kind of graphs.

A total Roman dominating function on a graph G is a function f: V (G)! f0; 1; 2g such that for every vertex v 2 V (G) with f(v) = 0 there exists a vertex u 2 V (G) adjacent to v with f(u) = 2, and the subgraph induced by the set fx 2 V (G): f(x)  1g has no isolated vertices. The total Roman DOMINATION number of G, denoted tR(G), is the minimum weight! (f) = P v2V (G) f(v) among all total Roman dominating functions f on G. It is known that tR(G)  t2(G) + (G) for any graph G with neither isolated vertex nor components isomorphic to K2, where t2(G) and (G) represent the semitotal DOMINATION number and the classical DOMINATION number, respectively. In this paper we give a constructive characterization of the trees that satisfy the equality above.

In this paper, we investigate DOMINATION number as well as signed DOMINATION numbers of Cay(G: S) for all cyclic group G of order n, where n ϵ {pm, pq} and S = {k < n: gcd(k, n) = 1}. We also introduce some families of connected regular graphs 􀀀 such that S (􀀀 ) ϵ {2, 3, 4, 5}.