Properties of Adjacency Matrix

Properties of Adjacency matrix

• if we raise the adjacency matrix to the power of L, the entry at position (m,n) of $A^L$ contains the number of unique walks of length L from node n to node m
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• Node indexing is arbitrary
• Permutation matrix
  • matrix where exactly one entry in each row and column take the value one, and the remaining values are zero
  • When position (m,n) of the permutation matrix is set to one, it indicates that node mwill become node nafter the permutation.
  • $X^{\prime }=XP;A^{\prime }=P^{T}AP$
  • post-multiplying by P permutes the columns and pre-multiplying by $P^T$ permutes the rows