For a connected graph Gthe characteristic polynomial of G is the determinant value of the matrix A(G)-λI, where A(G) is the adjacency of the matrix of G and I is the identity matrix. The roots of the characteristic polynomial equation are known as eigen values of G. The sum of the absolute values of the eigen values of G is called the energy of G and the largest eigen value is the spectral radius of G. Energies of molecular graphs have various applications in chemistry, polymerization, computer networking and pharmacy. In this paper we present the characteristic polynomial of certain graphs in terms of recurrence relation. Moreover we introduce a method to find the characteristic polynomial of a graph with single vertex deletion using adjacency Rhotrix.