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By h(G, x) and P (G, λ) we denote the adjoint polynomial and the chromatic polynomial of graph G, respectively. A new invariant of graph G, which is the fourth character R4(G), is given in this paper. Using the properties of the adjoint polynomials, the adjoint equivalence class of graph Bn−6,1,2 is determined, which can be regarded as the continuance of the paper written by Wang et al. [J. Wang, R. Liu, C. Ye and Q. Huang, A complete solution to the chromatic equivalence class of graph Bn−7,1,3, Discrete Math. (2007), doi: 10.1016/j.disc.2007.07.030]. According to the relations between h(G, x) and P (G, λ), we also simultaneously determine the chromatic equivalence class of Bn−6,1,2 that is the complement of Bn−6,1,2.

The chromatic equivalence class of graph Bn-6,12

The chromatic equivalence class of graph B_n-6,12