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Counting Coloured Graphs. III

Published online by Cambridge University Press:  20 November 2018

E. M. Wright
E. M. Wright*
Affiliation:
University of Aberdeen, Aberdeen, Scotland
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 1 , February 1972 , pp. 82 - 89
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Bellman, R., A brief introduction to theta-functions (Holt, Reinhart and Winston, New York, 1961).Google Scholar
2. Read, R. C., The number of k-coloured graphs on labelled nodes, Can. J. Math. 12 (1960), 409–413.Google Scholar
3. Read, R. C. and Wright, E. M., Coloured graphs: a correction and extension, Can. J. Math. 22 (1970), 594–596.Google Scholar
4. Wright, E. M., Counting coloured graphs, Can. J. Math. 13 (1961), 683–693.Google Scholar
5. Wright, E. M., Counting coloured graphs. II, Can. J. Math. 16 (1964), 128–135.Google Scholar