graph isomorphism

A graph isomorphism is a bijection between the vertices of two graphs $G$ and $H$:

$$f:V(G)\to V(H)$$

with the property that any two vertices $u$ and $v$ from $G$ are adjacent if and only if $f(u)$ and $f(v)$ are adjacent in $H$.

If an isomorphism can be constructed between two graphs, then we say those graphs are isomorphic.

For example, consider these two graphs:

Although these graphs look very different at first, they are in fact isomorphic; one isomorphism between them is

$$f(a)=1$$

$$f(b)=6$$

$$f(c)=8$$

$$f(d)=3$$

$$f(g)=5$$

$$f(h)=2$$

$$f(i)=4$$

$$f(j)=7$$