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Thread: Isomorphism Proof

  1. #1
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    Isomorphism Proof

    I'm having difficulty with this proof since I'm not sure how the two iffs affect the layout of the solution:

    Show that two simple graphs $\displaystyle G$ and $\displaystyle H$ are isomorphic if and only if there is a bijection $\displaystyle \vartheta : V(G) \rightarrow V(H)$ such that $\displaystyle uv \in E(G)$ if and only if $\displaystyle \vartheta (u) \vartheta (v) \in E(H)$.

    Any help would be much appreciated,
    Ultros
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  2. #2
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    It is considered impossible to prove a definition.
    If the above is not given as the definition of isomorphic graphs, then how does your textbook define isomorphic graphs?
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  3. #3
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    Defn of Isomorphic

    Two graphs G and H are said to be isomorphic if there are bijections $\displaystyle \theta : V(G) \rightarrow V(H)$ and $\displaystyle \phi : E(G) \rightarrow E(H)$ such that $\displaystyle \psi_G (e) = uv$ if and only if $\displaystyle \psi_H (\phi (e)) = \theta (u) \theta (v)$. Where $\displaystyle \psi$ is a graph's incidence function, e is an edge, and u and v are vertices.
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