I think the most common application is describing mesh connectivity in Finite Element Methods, where each entry in the matrix represents the influence each node has on each other. Basically any N^2 table can be constructed to describe the general dependency of components in any simulation or systems in genera.
Another notable example is the PageRank algorithm [0] where you consider the graph where nodes are web pages and edges are links between them and you can build an adjacency matrix of this graph and with this algorithm sort the pages based "popularity" (which pages have more links pointing to them intuitively)
Let's say that in most cases you have a graph and you consider the corresponding matrix. Doing the inverse is not as useful in practice except in some cases as explained in the article.
