Geodesic
Vertex and Edge Transitive, but not 1-Transitive, Graphs
Canadian mathematical bulletin, Tome 13 (1970) no. 2, pp. 231-237

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A (simple, undirected) graph G is vertex transitive if for any two vertices of G there is an automorphism of G that maps one to the other. Similarly, G is edge transitive if for any two edges [a, b] and [c, d] of G there is an automorphism of G such that {c, d} = {f(a), f(b)}. A 1-path of G is an ordered pair (a, b) of (distinct) vertices a and b of G, such that a and b are joined by an edge. G is 1-transitive if for any two 1-paths (a, b) and (c, d) of G there is an automorphism f of G such that c = f(a) and d = f(b). A graph is regular of valency d if each of its vertices is incident with exactly d of its edges. 
Bouwer, I. Z. Vertex and Edge Transitive, but not 1-Transitive, Graphs. Canadian mathematical bulletin, Tome 13 (1970) no. 2, pp. 231-237. doi: 10.4153/CMB-1970-047-8
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     title = {Vertex and {Edge} {Transitive,} but not {1-Transitive,} {Graphs}},
     journal = {Canadian mathematical bulletin},
     pages = {231--237},
     year = {1970},
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     number = {2},
     doi = {10.4153/CMB-1970-047-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1970-047-8/}
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