I would like to know if there is any class of graphs for which the adjacency matrices are invertible. At this moment I am aware of only the class of graphs $n K_2$ which is the disjoint union of $n$ number of $K_2$ matrices where the adjacency matrices are self-inverse.
Are there any other classes?
Given a permutation $\pi$ of a finite set $V$, form its cycle graph $G$ as follows: the vertex set is $V$ and the edges are pairs $(v,w)$ for which $\pi(v)=w$. (This is a simple directed graph.) The adjacency matrix will in fact be the permutation matrix corresponding to $\pi$, which is invertible.
We can also form graphs with loops whose adjacency matrices are upper triangular: take the vertex set $\{1,\cdots,n\}$ and adjoin edges $i\to j$ as one wishes but only when $i\le j$ (and of course make sure every vertex has at least one loop).