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An algebraic characterization of symmetric graphs with a prime number of vertices

Published online by Cambridge University Press:  17 April 2009

J.L. Berggren
J.L. Berggren
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada.
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A graph Γ is called symmetric if its automorphism group is transitive on its vertices and edges. Let p be an odd prime, Z(p) the field of integers modulo p, and Z*(p) = (a ∈ Z(p) | a ≠ 0}, the multiplicative subgroup of Z(p). This paper gives a simple proof of the equivalence of two statements:

(1) Γ is a symmetric graph with p vertices, each having degree n ≥ 1;

(2) the integer n is an even divisor of p − 1 and Γ is isomorphic to the graph whose vertices are the elements of Z(p) and whose edges are the pairs {a, a+h} where a ∈ Z(p) and h ∈ H, the unique subgroup of Z*(p) of order n.

In addition, the automorphism group of Γ is determined.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 7 , Issue 1 , August 1972 , pp. 131 - 134
Copyright
Copyright © Australian Mathematical Society 1972

References

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