What is the connection between adjacency matrix and topological measures What are the connections we can draw from a graph's adjacency matrix and topological measures that can be defined on the network induced by the graph? For example can we detect "holes" in a sensor network based on adjacency matrix akin to topological methods in this paper:
https://www2.math.upenn.edu/~ghrist/preprints/ipsn2005.pdf
It seems the adjacency matrix should contain information for many of the topological measures on a network/graph. Any references or insight would be greatly appreciated.

You can detect holes using much less.
If $G$ is a connected undirected graph, the number of holes (alternatively, the $\mathbb Z$-rank of the first homology group of $G$ considered as a CW-complex)  is just $ |E|- |V| + 1$.
You technically can get both $|V|$ and $|E|$ as the number of rows and half the sum of all entries of the adjacency matrix respectively.
