On Three Polynomials of Cyclic Silicate Networks

Abstract

Let a graph $G = (V, E)$ is a simple graph with vertex set V such that, $|V| =n.$ A $D(G, x)=\sum _{i=\gamma (G)}^{n}d(G, i)x^{i}$ is a domination polynomial of G, where $d(G, i)$ is the number of dominating sets of size i in G. ADi $(G, x)=\sum _{J}^{n} d_{i}(G, j)x^{J}$' is an independent domination polynomial of G, where $d_{i}(G, j)$ is the number of independent dominating sets of size j in G. $\mathrm {A} D_{t}(G, x)=\sum _{i=\gamma _{t}(G)}^{n}d_{t}(G, i)x^{i}$ is a total domination polynomial of G, where $d_{t}(G, i)$ is the number of total dominating sets of size i in G. In this work we studied $D(G, x)$, $D_{i}(G, x)$ and $D_{t}(G, x)$, and introduced some of their properties. Further, these polynomials for cyclic silicate network are computed.