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

Abstract

Let G be a graph with order n and G its complement. Denote by (G) the minimum real root of the adjoint polynomial of G. Two graphs G and H are chromatically equivalent if and only if G and H are adjointly equivalent. G 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 G with (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.