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At left is a graph, $G$, and at right is the Petersen graph.

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I think they're isomorphic. I would like to know if I'm correct and how I can start a proof.

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They aren't isomorphic, because the other graph has an obvious Hamiltonian cycle, but the Petersen graph is nonhamiltonian.

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No they are not isomorphic, since G has a loop with 4 vertices (C, J, G, D), and the Petersen graph has only loops of size 5. I drawed G in GeoGebra and moved the vertices around to get:

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