This question is a generalization of the question asked here.
From the answers of the questions, I can list four classes of graphs which have invertible adjacency matrices.
- The class of graphs $nK_2$
- Given a permutation $\pi$ of a finite set $V$, its cycle graph $G$ can be defined 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 be the permutation matrix corresponding to $\pi$, which is invertible. The class of all such graphs.
- Graphs with loops whose adjacency matrices are upper triangular: we 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 we have to make sure every vertex has at least one loop).
- Graphs with adjacency matrices which are diagonal with non-zero entries
Is there a systematic way to generate the group of all invertible adjacency matrices?