Geodesic
Locally arc-transitive graphs of valence $\{3,4\}$ with trivial edge kernel
Journal of Algebraic Combinatorics, Tome 38 (2013) no. 3, pp. 637-651.

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Summary: In this paper, we consider connected locally G-arc-transitive graphs with vertices of valence 3 and 4, such that the kernel $G_{uv}^{[1]}$ of the action of an edge-stabiliser on the neighbourhood $\Gamma (u)\cup \Gamma (v)$ is trivial. We find 19 finitely presented groups with the property that any such group G is a quotient of one of these groups. As an application, we enumerate all connected locally arc-transitive graphs of valence ${3,4}$ on at most 350 vertices whose automorphism group contains a locally arc-transitive subgroup G with $G_{uv}^{[1]} = 1$ .
Keywords: edge-transitive, locally arc-transitive, graph, symmetry, amalgam 
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     author = {Poto\v{c}nik, Primo\v{z}},
     title = {Locally arc-transitive graphs of valence $\{3,4\}$ with trivial edge kernel},
     journal = {Journal of Algebraic Combinatorics},
     pages = {637--651},
     publisher = {mathdoc},
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     number = {3},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2013__38_3_a4/}
}
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Potočnik, Primož. Locally arc-transitive graphs of valence $\{3,4\}$ with trivial edge kernel. Journal of Algebraic Combinatorics, Tome 38 (2013) no. 3, pp. 637-651. http://geodesic.mathdoc.fr/item/JAC_2013__38_3_a4/