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Properties of adjacency matrix of graph with cycle

  1. Dec 14, 2009 #1
    Let A be the adjacency matrix of some graph G. I am aware that A^n
    counts paths of length n between vertices of G, and that for graphs
    without cycles and non-singular A, (I-A)^-1 counts the total number of
    paths between vertices of G (correct me if any of this is wrong).This
    is a very limited class of graph however and I was wondering whether
    there is any useful information at all that can be obtained from the
    matrix (I-A)^-1 when A is non-singular and G contains a cycle (from
    the entries, determinant, etc.)? To take this further, what about if
    the matrix is singular? Is there any information that can be extracted
    other than counting paths of length n for each n?
    Thanks for any information.
  2. jcsd
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