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I didn't know where graph theory falls into, so I decided to post this question on here; please let me know if there is another thread where this post is more appropriate.

I am working on The Princeton Review's "Cracking the GRE Math Subject Test", and there was one problem about graph theory that I am puzzled about, and I was wondering if anyone can answer my question.

The question is the following:

Which of the following graphs are isomorphic?

Graph I:

Vertices are placed like:

BC

AD

and edges are: AB, BC, and CD

Graph II:

Vertices are placed like

FG

EH

and edges are: EG, GF, FH

Graph III:

J---K---L---M

(so edges are JK, KL, LM)

Sorry for a poor description of those graphs, but that's the best I could do. I thought all three of them were isomorphic, but the solution in the book says graphs I and II are not isomorphic by the following reason:

"Graph I has four vertices with the following edges: AB, BC, CD. Although there exists a bijective function f such that f(A) = E, f(B) = F, f(C) = G, and f(D) = H, adjacencies are not preserved; for example, there is no edge EF."

I'm quite confused about this argument because I thought f(A) = E, f(B) = G, f(C) = F, and f(D) = H is a bijective map that preserves adjacencies, and therefore I and II are isomorphic, but the solution in the book doesn't see this way.

If you know what is wrong, please let me know; it is possible that I'm not understanding the concept of isomorphisms of graphs.

Thanks!

PP

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# Isomorphisms of graphs

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