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I am trying to find a property (either similarity or isomorphism or other) that describe when two matrices are the same. More particularly, I am trying to describe adjacency matrices that are the same and derive graphs from it (as opposed to finding two graphs which are the same/isomorphic and derive adjacency matrices from it).

Ken
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tintinthong
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  • This is a link that gives a definition of isomorphism which looks very much like the definition of similar matrices. http://en.wikipedia.org/wiki/Adjacency_matrix (in properties section) – tintinthong Jun 09 '15 at 03:23
  • what do you two matrices are the same? do mean element by element? – abel Jun 09 '15 at 03:27
  • Agreed-- it is unclear what you mean by "the same." Also, the term "isomorphic" is not often used when describing matrices, nor is it used to describe matrices in the page you cite. – artificial_moonlet Jun 09 '15 at 03:29
  • Yes I have mistakenly read the sentence. It was written that the graphs were isomorphic and their respective adjacency matrices are similar. That is my mistake. – tintinthong Jun 09 '15 at 03:34

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The notion of isomorphism for adjacency matrices is that they are permutation similar, that is, they are similar via a permutation of the coordinates. This is stronger than similarity, which allows for any change of coordinates.

It is a simple result that two graphs are isomorphic if and only if their adjacency matrices are permutation similar, since an isomorphism of two graphs on the same number of vertices is exactly a bijective function respecting adjacency in both directions.

Here is an example of two graphs on five vertices that are not isomorphic, but have similar adjacency matrices. So we need this notion of permutation similarity; the standard linear algebra notion of similarity is too broad.

Community
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Andrew Dudzik
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