The chromatic equivalence classes of the complements of graphs with the minimum real roots of their adjoint polynomials greater than -4-4

Let GG be a graph with order nn and G¯ its complement. Denote by β(G)β(G) the minimum real root of the adjoint polynomial of GG. Two graphs GG and HH are chromatically equivalent if and only if G¯ and H¯ are adjointly equivalent. GG is chromatically unique if and only if G¯ is adjointly unique. In this paper, we give a method to determine all chromatic equivalence classes of a graph GG with β(G¯)>-4, by using some results on the minimum real roots of the adjoint polynomial of G¯. Moreover, we obtain a necessary and sufficient condition for those graphs that are chromatically unique.