Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-23T16:42:29.324Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymmetric and symmetric graphs

Published online by Cambridge University Press:  18 May 2009

E. M. Wright
E. M. Wright
Affiliation:
University of Aberdeen
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An (n, q) graph consists of n nodes and q edges, i.e. q distinct unordered pairs of different nodes, so that there are no loops or multiple edges. We write T for the number of unlabelled (n, q) graphs and F for the number of labelled (n, q) graphs. We say that a labelled graph is symmetric if there is a nonidentical permutation of its nodes which leaves the graph unaltered. We write r for the order of the automorphism group of the graph, i.e. the group of all those permutations of the nodes which leave the graph unaltered; we say that the graph is of symmetry order r. A graph which is not symmetric is called asymmetric and, for such a graph, obviously r = 1. We say that an unlabelled graph is symmetric or asymmetric according as the graph obtained by labelling its nodes is symmetric or asymmetric.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 15 , Issue 1 , March 1974 , pp. 69 - 73
Copyright
Copyright © Glasgow Mathematical Journal Trust 1974

References

REFERENCES

1.Erdös, P. and Renyi, A., Asymmetric graphs, Acta Math. Acad. Sci. Hungar. 14 (1963), 295–315, especially 298.CrossRefGoogle Scholar
2.Erdös, P. and Renyi, A., On random graphs I, Publ. Math. Debrecen 6 (1959), 290–297.Google Scholar
3.Euler, L., Solutio questionis curiosae ex doctrina combinationum, Mem. Acad. Sci. St. Petersbourg 3 (1811), 57–64; Opera omnia (1) 7 (1923), 435–448.Google Scholar
4.Polya, G., Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und Chemische Verbindungen, Acta Math. 68 (1937), 145–254.CrossRefGoogle Scholar
5.Riordan, J., An introduction to combinatorial analysis (New York, 1958), 57–62.Google Scholar
6.Wright, E. M., Graphs on unlabelled nodes with a given number of edges, Acta Math. 126 (1971), 1–9.CrossRefGoogle Scholar
7.Wright, E. M., The number of unlabelled graphs with many nodes and edges, Bull. Amer. Math. Soc. 78 (1972), 1032–1034.CrossRefGoogle Scholar