Without the word "connected" the answer is the disjoint union of a triangle and a square, per Smallest Graph that is Regular but not Vertex-Transitive?
I also know per wikipedia that the Frucht graph is a connected example with twelve vertices.
What is the smallest example of a connected regular graph which is not vertex-transitive?
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graph-theory
1 Answers
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The smallest example surely has $\leq \color{red}{8}$ vertices.
The depicted graph is a planar cubic graph, but is not vertex-transitive: there are three pentagonal faces meeting at the central point, and the central point is the only vertex with such a property. By a similar principle, here it is an example with $8$ vertices:
Some vertices belong to triangular faces, some don't.
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1Thanks! It remains to check 7 vertices, but this seems to be done in other postings - there is no 3-regular graph and there ar only very simple 2 and 4-regular graphs (http://math.stackexchange.com/questions/90492/number-of-non-isomorphic-regular-graphs-with-degree-of-4-and-7-vertices) – 2017-02-03
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1@SamuelCoskey: exactly, it should not be difficult to check that the minimal counter-examples have $8$ vertices. – 2017-02-03