I noticed that it is difficult to determine the necessary and sufficient conditions for a graph to have an invertible adjacency matrix (I've seen the other related posts on M.S.E.). I was wondering if there are any interesting applications of invertible adjacency matrices.
For example, is there any information encoded in the product of two adjacency matrices? An interesting result I have read is that the $(i,j)$ entry of $A^k$ for an adjacency matrix $A$ of a graph $G$ counts the number of walks from the vertex $v_i$ to $v_k$ that use $k$ edges.
