Graph isomorphism

In graph theory, an isomorphism of graphs (Isomorphic Graphs) G and H is a bijection between the vertex sets of G and H.

Note: In simple terms: Two graphs are isomorphic if they have same graph structure (edges) with the nodes simply renamed (re-labelled).

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

Such that any two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H.

This kind of bijection is commonly described as "edge-preserving bijection", in accordance with the general notion of isomorphism being a structure-preserving bijection.

Isomorphic graphs are denoted as: $G\simeq H$

In the case when the bijection is a mapping of a graph onto itself, i.e., when G and H are one and the same graph, the bijection is called an automorphism of G.

The two graphs shown below are isomorphic, despite their different looking drawings.

Graph Isomorphism

Graph G	Graph H	An isomorphism between G and H
		$f(a)=1$
		$f(b)=6$
		$f(c)=8$
		$f(d)=3$
		$f(g)=5$
		$f(h)=2$
		$f(i)=4$
		$f(j)=7$