An automorphism is an isomorphism of a mathematical object or system of objects onto itself.

The automorphism group of a complete bipartite graph K_n,n is (S_n x S_n) semidirect Z_2.

Yes!

Frucht Theorem: Each finite group is realized as full automorphism group of a graph. The proof is constructive, so you can obtain your graph For instance: add 5-rays of different length to a 5-cycle.

Pierre-Emmanuel Caprace has written:

'\\' -- subject(s): Isomorphisms (Mathematics), Automorphism, Kac-Moody algebras