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New families of adjointly unique graphs. (English) Zbl 0883.05048
The adjoint polynomial $$h(G,x)$$ of a graph $$G$$ is derived from the chromatic polynomial of its complement. The graph $$G$$ is called adjointly unique if $$h(H,x)=h(G,x)$$ implies that $$H$$ is isomorphic with $$G$$. It is clear that a graph is adjointly unique if and only if its complement is chromatically unique. In the paper is presented a class of adjointly unique graphs, which is formed from paths, cycles and $$D_n$$ graphs. $$D_n$$ is a graph obtained from the union of $$P_{n-2}$$ and $$K_3$$ by identifying a vertex from $$K_3$$ with one endvertex of $$P_{n-2}$$.
##### MSC:
 05C15 Coloring of graphs and hypergraphs