Let A be the adjacency matrix of some graph G. I am aware that A^n(adsbygoogle = window.adsbygoogle || []).push({});

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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# Properties of adjacency matrix of graph with cycle

Can you offer guidance or do you also need help?

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