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If $G$ is a vertex-transitive graph then it is well known that the edge connectivity of $G$, $\kappa'(G)$ equals to $\rm{val}(G)$ - the degree of a vertex in $G$ (note that $G$ is regular.) My question is the following

Is $\kappa'(G) = \delta(G)$ for every edge-transitive graph $G$ that is not necessarily vertex-transitive?

The claim appears to be true after testing it on some of the known edge-transitive graphs (that are not vertex-transitive). However I do not see an obvious way to modify the proof for vertex-transitive graphs to make it work for this case as well.

Is the claim true? Could someone give a hint for its proof ?

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I suspect that edge-connectivity does equal minimum valency for edge-transitive graphs. I also suspect that the usual proof of this for vertex-transitive graphs can be extended to this case. Since this is a nice question, I think I will assign it to my graduate class, and so I will not be more explicit here. –  Chris Godsil Jan 24 '13 at 20:06

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