For a graph G, let $A( G )$ be its adjacency matrix. Let $\varphi _G ( x )$ be the characteristic polynomial of G. Let J be a matrix with all entries equal to 1. Let $\psi _G ( x ) = \varphi_{A ( G ) - J} ( x ) - \varphi_G ( x )$. In this paper, we show that the haracteristic polynomials of the join $G + H$, the complement $\bar G$ and the composition $G [ H ]$ can be expressed in terms of $\varphi _G ,\varphi _H ,\psi _G $ and $\psi _H $.