Is there a way to make two different adjacency matrices (same size) have the same degree matrix?

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For the same set of vertices, there are two different topologies and edge weights. Therefore, we have two different adjacency matrices for the same vertices. Is it possible to make those two different adjacency matrices have the same degree matrix?

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It is certainly possible. Note that the idea of a degree matrix can be represented more concisely by a degree sequence, since a degree matrix is diagonal.

For an explicit example of two graphs with different adjacency matrices but the same degree matrix, consider:

  • the 1-skeleton of the pentagonal prism (its vertices and edges), which has girth 4
  • the Petersen graph of girth 5

Both of them have different adjacency matrices, but the same degree matrix of three times the identity matrix of size 10 (i.e. a degree sequence of ten threes).