A way to construct independence equivalent graphs

Let us denote the independence polynomial of a graph by I G ( x ) . If I G ( x ) = I H ( x ) implies that G ≅ H then we say G is independence unique. For graph G and H if I G ( x ) = I H ( x ) 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.