Chromatic equivalence classes of certain cycles with edges

Let P(G) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent, written G∼H, if P(G)=P(H). A graph G is chromatically unique if for any graph H, G∼H implies that G is isomorphic with H. In this paper, we give the necessary and sufficient conditions for a family of generalized polygon trees to be chromatically unique.