Abstract: Two graphs are chromatically equivalent if they have the same chromatic polynomials. A graph is said to be chromatically unique if each graph which has same chromatic polynomial is isomorphic with it. Similarly, two graphs are adjointly equivalent if they have the same adjoint polynomials. A graph is said to be adjointly unique if each graph which has same adjoint polynomial is isomorphic with it. As we all know, two graphs are chromatically equivalent if their complement are adjointly equivalent; a graph is chromatically unique if its complement is adjointly unique. In this paper, a necessary and sufficient condition of a classe graphs adjointly unique is given, thus a necessary and sufficient condition of their complement chromatically unique is given.

Key words: chromatic polynomial, adjoint polynomial, chromatic uniquness

摘要： 如果两个图的色多项式相等，称这两个图色等价。如果与一个图色等价的所有图都与这个图同构，称这个图色唯一。类似的，如果两个图的伴随多项式相等，称这两个图伴随等价。如果与一个图伴随等价的所有图都与这个图同构，称这个图伴随唯一。众所周知，两个图色等价当且仅当它们的补图伴随等价；一个图色唯一当且仅当它的补图伴随唯一。给出了一类图伴随唯一的一个充分必要条件，因而给出了它的补图色唯一的一个充分必要条件。

关键词: 色多项式, 伴随多项式, 色唯一, 伴随唯一