Let us consider two bangles. On these bangles let us add $8$ pearls on each bangle, the sequence of colors of pearls are same on both bangle. But the sequence of labels on pearls is not same. Now assuming the pearls as vertices of graph and circumference of bangle as edges of a graph, these bangles would correspond to two undirected graph, say $G_1$ and $G_2$ .
As far as I think these two graphs would be isomorphic as their number of vertices, edges and connectedness and other properties are same. Additionally, I cannot think of a property that is not same in both. Am I right?
Graph isomorphism means that there exist a bijection $f$ such that $f:V(G_1) \to V(G_2)$. So, this bijection must be a function. The question is does such a function means that we can define a formula to get vertex of $G_2$ as output when the input is a vertex of $G_1$. (Obviously it should preserve the edges)
Is finding such an $f$ equals to graph isomorphism problem.