Alikhani, S., Jafari, N. (2018). TOTAL DOMINATION POLYNOMIAL OF GRAPHS FROM PRIMARY SUBGRAPHS. Journal of Algebraic Systems, 5(2), 127-138. doi: 10.22044/jas.2018.1096

S. Alikhani; N. Jafari. "TOTAL DOMINATION POLYNOMIAL OF GRAPHS FROM PRIMARY SUBGRAPHS". Journal of Algebraic Systems, 5, 2, 2018, 127-138. doi: 10.22044/jas.2018.1096

Alikhani, S., Jafari, N. (2018). 'TOTAL DOMINATION POLYNOMIAL OF GRAPHS FROM PRIMARY SUBGRAPHS', Journal of Algebraic Systems, 5(2), pp. 127-138. doi: 10.22044/jas.2018.1096

Alikhani, S., Jafari, N. TOTAL DOMINATION POLYNOMIAL OF GRAPHS FROM PRIMARY SUBGRAPHS. Journal of Algebraic Systems, 2018; 5(2): 127-138. doi: 10.22044/jas.2018.1096

TOTAL DOMINATION POLYNOMIAL OF GRAPHS FROM PRIMARY SUBGRAPHS

^{1}Department of Mathematics, Yazd University, 89195-741, Yazd, Iran.

^{2}Department of Mathematics, Yazd University, 89195-741 Yazd, Iran.

Abstract

Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set 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 denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum d_t(G,i)$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. Let $G$ be a connected graph constructed from pairwise disjoint connected graphs $G_1,\ldots ,G_k$ by selecting a vertex of $G_1$, a vertex of $G_2$, and identify these two vertices. Then continue in this manner inductively. We say that $G$ is obtained by point-attaching from $G_1, \ldots ,G_k$ and that $G_i$'s are the primary subgraphs of $G$. In this paper, we consider some particular cases of these graphs that most of them are of importance in chemistry and study their total domination polynomials.