Geodesic
Counting Coloured Graphs. III
Canadian journal of mathematics, Tome 24 (1972) no. 1, pp. 82-89

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In an earlier paper [4], we found an asymptotic expansion for Mn = Mn(k), the number of coloured graphs on n labelled nodes, when n is large. Such a graph is a set of n distinguishable objects called nodes, and a set of “edges”, that is, undirected pairs of nodes. The nodes are mapped onto k colours. Every pair of nodes of different colours may or may not be joined by an edge, but no edge can join a pair of nodes of the same colour.We write mn for the number of these graphs which are connected, Fn for the number which use all k colours (i.e., at least one node in each graph is mapped onto each of the k colours), and fn for the number of connected graphs which use all k colours. 
Wright, E. M. Counting Coloured Graphs. III. Canadian journal of mathematics, Tome 24 (1972) no. 1, pp. 82-89. doi: 10.4153/CJM-1972-010-7
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[1] 1. Bellman, R., A brief introduction to theta-functions (Holt, Reinhart and Winston, New York, 1961). Google Scholar

[2] 2. Read, R. C., The number of k-coloured graphs on labelled nodes, Can. J. Math. 12 (1960), 409–413. Google Scholar

[3] 3. Read, R. C. and Wright, E. M., Coloured graphs: a correction and extension, Can. J. Math. 22 (1970), 594–596. Google Scholar

[4] 4. Wright, E. M., Counting coloured graphs, Can. J. Math. 13 (1961), 683–693. Google Scholar

[5] 5. Wright, E. M., Counting coloured graphs. II, Can. J. Math. 16 (1964), 128–135. Google Scholar

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