Node ordering permutation based adjacency matrix 
An undirected graph $G=(V, E)$ is defined by its node set
$V=\left\{v_1, \ldots, v_n\right\}$ and edge set $E=\left\{\left(v_i,
> v_j\right) \mid v_i, v_j \in V\right\}$. One common way to represent a
graph is using an adjacency matrix, which requires a node ordering
$\pi$ that maps nodes to rows/columns of the adjacency matrix. More
precisely, $\pi$ is a permutation function over $V$ (i.e.,
$\left(\pi\left(v_1\right), \ldots, \pi\left(v_n\right)\right.$ ) is a
permutation of $\left.\left(v_1, \ldots, v_n\right)\right)$. We define
$\Pi$ as the set of all $n$ ! possible node permutations. Under a node
ordering $\pi$, a graph $G$ can then be represented by the adjacency
matrix $A^\pi \in \mathbb{R}^{n \times n}$, where $A_{i,
> j}^\pi=\mathbb{1}\left[\left(\pi\left(v_i\right),
> \pi\left(v_j\right)\right) \in E\right]$.
Note that elements in the set of adjacency matrices $A^{\Pi}=$
$\left\{A^\pi \mid \pi \in \Pi\right\}$ all correspond to the same
underlying graph.

I was reading a research paper where I came across this definition of an adjacency matrix based on a node ordering function. I am a beginner in graph theory hence was not able to understand the concept of node permutation. Any help is appreciated.

*

*paper: https://arxiv.org/abs/1802.08773 GraphRNN by Stanford CS.

*Quora: https://www.quora.com/unanswered/What-does-the-adjacency-matrix-for-permutation-mean-in-the-GraphRNN-paper

*reddit: https://www.reddit.com/r/learnmachinelearning/comments/nx06n0/what_does_the_adjacency_matrix_for_permutation/
 A: Edit in response to the question in the bounty.
The nodes in the graph come you you in some particular order - they are numbered from $1`$ to $n$. $A^\pi_{i,j}$ is the adjacency matrix for the graph when you list the vertices in the order determined by the permutation $\pi$ of the sequence $\{1,2,\ldots,n\}$.
The meaning of the adjacency matrix is independent of $\pi$, which does nothing to the graph.

@dan_fulea has written a correct answer that may be more elaborate than what you need. This part of his last paragraph is what I think you are asking about:

The cited text is very pedant, puts the accent on some bureaucratic
point, the order, then introduces a lot of notation for this,

For example, consider the graph with vertices $A,B,C$ and the single edge $AB$. If you write the vertices in that order tha adjacency matrix is
0 1 0
1 0 0
0 0 0

while if you write the nodes in order $A,C,B$ the same information is encoded in the matrix
0 0 1
0 0 0
1 0 0 

The long discussion of permutations just makes this obvious fact precise.
