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On the number of simple eigenvalues of a graph

Published online by Cambridge University Press:  14 November 2011

Peter Rowlinson
Peter Rowlinson
Affiliation:
Department of Mathematics, University of Stirling, Stirling, Scotland
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Synopsis

Let Γ be a graph with n points, and let G be the group of automorphisms of Γ. An orbit of G on which G acts as an elementary abelian 2-group is said to be exceptional. It is shown that the number of simple eigenvalues of Γ is at most (5n+4t)/9, where t is the number of points of Γ lying in exceptional orbits of G.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 94 , Issue 3-4 , 1983 , pp. 247 - 250
Copyright
Copyright © Royal Society of Edinburgh 1983

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References

1.Babai, L.. Automorphism group and category of cospectral graphs. Acta Math. Acad. Sci. Hungar. 31 (1978), 295–306.CrossRefGoogle Scholar
2.Biggs, N. L.. Algebraic Graph Theory (London: Cambridge University Press, 1974).CrossRefGoogle Scholar
3.Cvetkovic, D. M., Doob, M. and Sachs, H.. Spectra of Graphs (New York: Academic Press, 1980).Google Scholar
4.Gorenstein, D.. Finite Groups (New York: Harper and Row, 1968).Google Scholar
5.Mowshowitz, A.. The group of a graph whose adjacency matrix has all distinct eigenvalues. Proof Techniques in Graph Theory (Harary, F., editor), pp. 109–110 (New York: Academic Press, 1969).Google Scholar
6.Petersdorf, M. and Sachs, H.. Spektrum and Automorphismengruppe eines Graphen. Combinatorial Theory and its Applications, Vol. III (Erdós, P., Rényi, A. and Sós, Vera T., editors), pp. 891–907 (London: North Holland, 1970).Google Scholar
7.Rodemich, E.. The groups of order 128. J. Algebra 67 (1980), 129–142.CrossRefGoogle Scholar
8.Sachs, H. and Stiebitz, M.. Automorphism group and spectrum of a graph. Algebraic Methods in Graph Theory, Vol. II (Lovász, L. and Sós, Vera T., editors), pp. 657–670 (Amsterdam: North Holland, 1981).Google Scholar
9.Sachs, H. and Stiebitz, M.. Konstruktion schlichter transitiver Graphen mit maximaler Anzahl einfacher Eigenwerte. Math. Nachr. 100 (1981), 145–150.Google Scholar
10.Wielandt, H.. Finite Permutation Groups (New York: Academic Press, 1964).Google Scholar