Half-Transitive Graphs of Prime-Cube Order |
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Authors: | Ming-Yao Xu |
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Institution: | (1) Institute of Mathematics, Peking University, Beijing, 100871, China |
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Abstract: | We call an undirected graph X half-transitive if the automorphism group Aut X of X acts transitively on the vertex set and edge set but not on the set of ordered pairs of adjacent vertices of X. In this paper we determine all half-transitive graphs of order p
3 and degree 4, where p is an odd prime; namely, we prove that all such graphs are Cayley graphs on the non-Abelian group of order p
3 and exponent p
2, and up to isomorphism there are exactly (p – 1)/2 such graphs. As a byproduct, this proves the uniqueness of Holt's half-transitive graph with 27 vertices. |
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Keywords: | half-transitive graphs Cayley graphs simple groups Schur multiplier |
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