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Question

# Show that isomorphism of simple graphs is an equivalence relation.

Solution

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DEFINITIONS

Two simple graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ are $\textbf{isomorphic}$ if there exists a one-to-one and onto function $f:V_1\rightarrow V_2$ such that $a$ and $b$ are adjacent in $G_1$ if and only if $f(a)$ and $f(b)$ are adjacent in $G_2$.

A relation $R$ is an $\textbf{equivalence relation}$ if the relation $R$ is transitive, symmetric and reflexive.

A relation $R$ on a set $A$ is $\textbf{reflexive}$ if $(a,a)\in R$ for every element $a\in A$.

A relation $R$ on a set $A$ is $\textbf{symmetric}$ if $(b,a)\in R$ whenever $(a,b) \in R$

A relation $R$ on a set $A$ is $\textbf{transitive}$ if $(a,b)\in R$ and $(b,c) \in R$ implies $(a,c)\in R$

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