A Complete Solution to the Chromatic Equivalence Class of Graph \(\overline{\zeta_n^1}\)

Two graphs are defined to be adjointly equivalent if their complements are chromatically equivalent. By \( h(G,x) \) and \( P(G,\lambda) \) we denote the adjoint polynomial and the chromatic polynomial of graph \( G \), respectively. A new invariant of graph \( G \), which is the fifth character \( R_5(G) \), is given in this paper. Using this invariant and the properties of the adjoint polynomials, we firstly and completely determine the adjoint equivalence class of the graph \( \zeta_n^1 \). According to the relations between \( h(G,x) \) and \( P(G,\lambda) \), we also simultaneously determine the chromatic equivalence class of \( \overline{\zeta_n^1} \).