A graph isomorphism is a bijection between the vertices of two graphs $G$ and $H$ $$f: V(G) \rightarrow V(H)$$ 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:

$$\xymatrix{ a \ar@{-}[r] \ar@{-}[rd] \ar@{-}[rdd] & g \\ b \ar@{-}[ru] \ar@{-}[r] \ar@{-}[rdd] & h \\ c \ar@{-}[ruu] \ar@{-}[r] \ar@{-}[rd] & i \\ d \ar@{-}[ruu] \ar@{-}[ru] \ar@{-}[r] & j}$$

$$\xymatrix{ 1 \ar@{-}[rrr] \ar@{-}[rd] \ar@{-}[ddd] & & & 2 \\ & 5 & 6 \ar@{-}[ur] \ar@{-}[l] \ar@{-}[d] & \\ & 8 \ar@{-}[u] \ar@{-}[r] \ar@{-}[dl] & 7 & \\ 4 & & & 3 \ar@{-}[uuu] \ar@{-}[lll] \ar@{-}[ul] }$$

Although these graphs look very different at first, they are in fact isomorphic; one isomorphism between them is $$f(a) = 1$$ $$f(b) = 6$$ $$f(c) = 8$$ $$f(d) = 3$$ $$f(g) = 5$$ $$f(h) = 2$$ $$f(i) = 4$$ $$f(j) = 7$$