Geodesic
The vertex-transitive and edge-transitive tetravalent graphs of square-free order
Journal of Algebraic Combinatorics, Tome 42 (2015) no. 1, pp. 25-50.

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In this paper, a classification is given for tetravalent graphs of square-free order which are vertex-transitive and edge-transitive. It is shown that such graphs are Cayley graphs, edge-regular metacirculants and covers of some graphs arisen from simple groups $\mathrm A_7$, $\mathrm J_1$ and $\mathrm{PSL}(2,p)$.
Classification : 05C25
Keywords: edge-transitive graph, vertex-transitive graph, arc-transitive graph, coset graph, automorphism group 
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Li, Cai Heng; Lu, Zai Ping; Wang, Gai Xia. The vertex-transitive and edge-transitive tetravalent graphs of square-free order. Journal of Algebraic Combinatorics, Tome 42 (2015) no. 1, pp. 25-50. http://geodesic.mathdoc.fr/item/JAC_2015__42_1_a10/