# Is this a sufficient condition for vertex and edge transitivity?

I'm traying to prove (or disprove) the following statement:

Any connected $r$-regular graph of girth $g$ such that every edge is shared by the same number of minimum length cycles (that is, cycles of length $g$), is vertex-transitive and edge-transitive.

This is not a textbook exercise. Any ideas appreciated.

Thanks.

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The statement is false if the graph can be disconnected. A connected (4,4)-graph can be constructed by drawing a grid on the torus. Consider the disjoint union of two such graphs with different numbers of vertices. – JeffE May 13 '12 at 16:13
@JeffE Of course you're right. I corrected the question. – becko May 13 '12 at 18:06