Abstract: Let S_n be the star with n vertices, and let G be any connected graph with p vertices. We denote by E_rp+(r-1) the graph obtained from S_r and rG by coinciding the i-th vertex of G with the vertex of degree r-1 of S_r, while the i-th vertex of each component of (r-1)G be adjacented to r-1 vertices of degree 1 of S_r, respectively. By applying the properties of adjoint polynomials. We prove that factorization theorem of adjoint polynomials of kinds of graphs Erp+(r-1) (r-1)K1 (1 ≤ i ≤ p). Furthermore, we obtain structure characteristics of chromatically equivalent graphs of their complements. 

Key words: chromatic polynomial, adjoint polynomials, factorization, chromatically equivalent graph, structure characteristics