A complete solution to the chromatic equivalence class of graph Bn-7,1,3¯

By h(G,x)h(G,x) and P(G,λ)P(G,λ) we denote the adjoint polynomial and the chromatic polynomial of a graph G, respectively. A new invariant of graph G  , which is the fourth character R4(G)R4(G), is given. By the properties of the adjoint polynomials, the adjoint equivalence class of graph Bn-7,1,3Bn-7,1,3 is determined. According to the relations between h(G,x)h(G,x) and P(G,λ)P(G,λ), we also simultaneously determine the chromatic equivalence class of Bn-7,1,3¯ which is the complement of Bn-7,1,3Bn-7,1,3.