# Question about Isomorphism with Subgraphs

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.

• There is no essential difference between the two. The vertices may be labelled differently, that's all. Sep 1 '20 at 0:07

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 $$H' = f(H)$$.