The chromatic polynomial between graph \& its complement---about Akiyama and Harary's open problem (Q1906858)

\textit{J. Akiyama} and \textit{F. Harary} proposed the following problem [The theory and applications of graphs, 4th int. Conf., Kalamazoo/Michigan 1980, 1-12 (1981; Zbl 0468.05023)]: ``Are there any graphs \(G\) which are not self-complementary with \(G\) and \(\overline G\) having the same chromatic polynomial?'' In this paper it is shown that there are no such graphs \(G\) when \(|V(G)|= p< 8\) or \(p\equiv 2, 3\pmod 4\). Furthermore, there exists at least one graph \(G\) of order \(p\) which is not self-complementary such that \(G\) and \(\overline G\) have the same chromatic polynomial for every \(p\equiv 0, 1\pmod 4\) and \(p\geq 8\). The existence proof is by construction.