A Complete Solution to the Chromatic Equivalence Class of Graph $\overline{{\zeta }_n^1}$

Abstract

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 $\overline{{\zeta }_n^1}$.