The enumeration of self-complementary \(k\)-multigraphs (Q1582580)

In a \(k\)-multigraph \(G\), any pair of vertices can be joined by at most \(k\) edges. If every pair of vertices joined by \(i\) edges are instead joined by \(k-i\) edges, then \(G\) is transformed into its complement \(\overline G\). If \(G\) is isomorphic to \(\overline G\), then \(G\) is self-complementary. The paper under review uses Pólya theory to obtain a formula for the number of non-isomorphic self-complementary \(k\)-multigraphs with \(p\) vertices. This result generalizes to \(k\)-multigraphs an analogous result for graphs in \textit{R. C. Read} [J. Lond. Math. Soc. 38, 99-104 (1963; Zbl 0116.15001)].