How is the permutation done in this example?

Question

I have confusion in understanding how the permutation is done in this example.

Answer 1

The permutation $\phi$ is being written in cyclic notation, so you interpret $\phi = (1, 2, 4, 3)$ as meaning $\phi(1) = 2, \phi(2) = 4, \phi(4) = 3, \phi(3) = 1, \phi(5) = 5$ - within the cycle, each element is mapped to the next one in the list (wrapping around when you get to the end), and anything not listed in the cycle stays unchanged.

If we just relabel the nodes in the graph without changing their position, the diagram looks like this:

Then to get from the right-hand side of this image to the one you've posted, you just have to move the nodes back to their original position.