Enumeration of mixed graphs
Frank Harary, Ed Palmer · Proceedings of the American Mathematical Society · 1966
and three oriented lines. An ordinary graph may be regarded as a mixed graph with no oriented lines, and an oriented graph as a mixed graph with no ordinary lines. Further, any digraph may be considered as a mixed graph by changing each symmetric pair of lines to an ordinary line. Our object is to derive a formula which enumerates mixed graphs on p points with respect to the number of ordinary and oriented lines. For graphical definitions we refer to [4], [5]. Let mpqr be the number of mixed graphs with p points having exactly q oriented lines and r ordinary lines. Then the polynomial m,(x, y) which enumerates mixed graphs with p points according to both the number of ordinary and oriented lines is defined by