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If two graphs are isomorphic does that mean that all possible subgraphs of adjacent vertices from a certain vertex from both graphs must be isomorphic?

Also is it all possible subgraphs? Or all possible corresponding subgraphs of adjacent vertices from a certain vertex from both graphs must be isomorphic? What is the difference between the two?

Thank You

I wish to develop my understanding of graph theory.

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  • $\begingroup$ There is no essential difference between the two. The vertices may be labelled differently, that's all. $\endgroup$
    – saulspatz
    Sep 1 '20 at 0:07
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If you have an isomorphism, then everything about them is the same. In particular, if we call our graphs $ G $ and $ G' $, and $ f : G \rightarrow G' $ is the isomorphism, then I leave you to check that if $ H $ is any subgraph of $ G $, then $ H $ will be isomorphic to $ H' = f(H) $.

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