Geodesic
Half-transitive graphs of prime-cube order
Journal of Algebraic Combinatorics, Tome 1 (1992) no. 3, pp. 275-282.

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Summary: 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.
Keywords: half-transitive graphs, Cayley graphs, simple groups, Schur multiplier 
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Xu, Mingyao. Half-transitive graphs of prime-cube order. Journal of Algebraic Combinatorics, Tome 1 (1992) no. 3, pp. 275-282. http://geodesic.mathdoc.fr/item/JAC_1992__1_3_a1/