This question originates from my “revelation” that the identity matrix generates the world's most boring graph: all the nodes are just connected to themselves and none of them are connected to each other.
Given an adjacency matrix $A$ for a graph $G$, what do things like the determinant and eigenvalues / eigenvectors of $A$ really tell us about $G$?
If $A$ forms an orthogonal basis, then $G$ is quite “boring”, as I said. So, clearly, for “interesting” graphs, no inner product will form in the vector space defined by $A$. But $A$ still has a unique determinant, eigenvalues and eigenvectors, a inverse, a minimal polynomial and characteristic polynomial, and a Jordan normal form. What do the characteristics of these properties of $A$ mean for $G$, or even $G^n$?
I understand these are likely loaded questions, but I want to find a clearer link from Graph Theory to Linear Algebra.