example of graph such that deleting any edge couses lack of automorphism

Question

Could you try give an example graph, such that deleting any edge causes lack of automorphism ? Graph before deleting operation should have automorphism.

The only thing that I can find is examaple, but I choose edge to delete, but when it comes to deleting any edge I am helpless.

Answer 1

If you restrict yourself to graphs of minimum degree $\ge$ 1 then this is the smallest example