Review of functions:

A function $f:A\rightarrow B$ is one-to-one, if no two elements of $A$ have the same image. So if $a\not=a'$, then $f(a)\not=f(a')$. In other words, every element of $B$ has at most one pre-image.

A function $f:A\rightarrow B$ is onto, if every element of $B$ has at least one pre-image. So for every element $b\in B$, there exists an element $a\in A$ so that $f(a)=b$.

A function $f:A\rightarrow B$ is a bijection if it is one-to-one and onto. Note that this implies that every element in $B$ has exactly one pre-image.

For a bijection we can form the inverse function $f^{-1}:B\rightarrow A$, where $f^{-1}(b)=a$ precisely when $f(a)=b$.

If $f:A\rightarrow B$ is a bijection, then $\vert A\vert=\vert B\vert$

Isomorphism

An (undirected) graph $G_1=(V_1,E_1)$ is isomorphic to another graph $G_2=(V_2,E_2)$ if there exists a bijection $f:V_1\rightarrow V_2$ so that, for all $a,b\in V_1$,

$$\{a,b\}\in E_1 \Leftrightarrow \{f(a),f(b)\}\in E_2.$$

The bijection $f$ is called a graph isomorphism.

Similarly:

A directed graph $G_1=(V_1,E_1)$ is isomorphic to another digraph $G_2=(V_2,E_2)$ if there exists a bijection $f:V_1\rightarrow V_2$ so that, for all $a,b\in V_1$,

$$(a,b)\in E_1 \Leftrightarrow (f(a),f(b))\in E_2.$$