In this paper, our research primarily focuses on f-symmetric and unimodal properties of independence polynomials \(I(G\circ (K_{p}\cup K_{q});x)\) . More precisely, we prove that the independence polynomial \(I(G\circ (K_{p}\cup K_{q});x)\) is unimodal for every graph G, whenever the positive integers p and q are large enough. In addition, we establish several inequalities involving the coefficients of \(I(G\circ (K_{p}\cup K_{q});x)\) for arbitrary p and q.
