Properties of adjacency matrix of graph with cycle

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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.
 

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