The calling sequence PermuteVertices('G','sigma') returns a new graph H with Vertices(H) = sigma.  The list of neighbors data structure is reordered according to sigma so that the adjacency matrix of H will be different in general.  Attribute information, including vertex position information is also permuted according to sigma so that DrawGraph(H) will look identical to DrawGraph(G).

The calling sequence IsomorphicCopy('G','sigma') returns a new graph H where the list of neighbors data structure is reordered according to sigma but the vertex labels of H are the same as G. It also discards all attributes from G so that if H is drawn, it will not be obvious that H is isomorphic to G.

The calling sequence PermuteVertices('G') chooses a random permutation sigma of the vertices of G then returns H = PermuteVertices(G,sigma). Hence Vertices(H) is the permutation used.

The calling sequence IsomorphicCopy('G') chooses a random permutation sigma of the vertices of G and returns IsomorphicCopy('G','sigma').

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$\mathrm{with}\left(\mathrm{SpecialGraphs}\right)\:$

$G≔\mathrm{PathGraph}\left(5\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 4\; edge(s)}$

$\mathrm{Vertices}\left(G\right),\mathrm{Neighbors}\left(G\right)$

$\left[1\,2\,3\,4\,5\right],\left[\left[2\right]\,\left[1\,3\right]\,\left[2\,4\right]\,\left[3\,5\right]\,\left[4\right]\right]$

$H≔\mathrm{PermuteVertices}\left(G\,\left[3\,5\,1\,2\,4\right]\right)$

$H≔\mathrm{Graph\; 2:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 4\; edge(s)}$

$\mathrm{Vertices}\left(H\right)$

$\left[3\,5\,1\,2\,4\right]$

$\mathrm{Neighbors}\left(H\right)$

$\left[\left[2\,4\right]\,\left[4\right]\,\left[2\right]\,\left[3\,1\right]\,\left[3\,5\right]\right]$

$H≔\mathrm{IsomorphicCopy}\left(G\,\left[3\,5\,1\,2\,4\right]\right)$

$H≔\mathrm{Graph\; 3:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 4\; edge(s)}$

$\mathrm{Vertices}\left(H\right),\mathrm{Neighbors}\left(H\right)$

$\left[1\,2\,3\,4\,5\right],\left[\left[4\,5\right]\,\left[5\right]\,\left[4\right]\,\left[1\,3\right]\,\left[1\,2\right]\right]$

$H≔\mathrm{PermuteVertices}\left(G\right)$

$H≔\mathrm{Graph\; 4:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 4\; edge(s)}$

$\mathrm{\sigma }≔\mathrm{Vertices}\left(H\right)$

$\mathrm{\sigma }≔\left[3\,4\,5\,1\,2\right]$

$P≔\mathrm{PrismGraph}\left(3\,3\right)$

$P≔\mathrm{Graph\; 5:\; an\; undirected\; graph\; with\; 6\; vertices\; and\; 9\; edge(s)}$

$H≔\mathrm{IsomorphicCopy}\left(P\,\left[4\,1\,2\,6\,5\,3\right]\right)$

$H≔\mathrm{Graph\; 6:\; an\; undirected\; graph\; with\; 6\; vertices\; and\; 9\; edge(s)}$

$\mathrm{DrawGraph}\left(P\right)$

$\mathrm{DrawGraph}\left(H\,\mathrm{style}=\mathrm{spring}\right)$