A way to construct independence equivalent graphs

Let us denote the independence polynomial of a graph by $I_G\left(x\right)$. If $I_G\left(x\right)=I_H\left(x\right)$ implies that $G?H$ then we say $G$ is independence unique. For graph $G$ and $H$ if $I_G\left(x\right)=I_H\left(x\right)$ but $G$ and $H$ are not isomorphic, then we say $G$ and $H$ are independence equivalent. In 7], Brown and Hoshino gave a way to construct independent equivalent graphs for circulant graphs. In this work we give a way to construct the independence equivalent graphs for general simple graphs and obtain some properties of the independence polynomial of paths and cycles.