Which are the sufficient and necessary conditions for an undirected graph with no self edges (i.e. no loop of length $1$) to have an invertible adjacency matrix?
In this case, the adjacency matrix is symmetric (i.e. $A = A^\top$). Moreover, all the diagonal elements are $0$ and there is a $1$ in both the entries $(i,j)$ and $(j,i)$, with $i\neq j$, if and only if vertices $i$ and $j$ are connected.
A first necessary condition is the following: all vertices must have at least one connection, otherwise the relative row of $A$ is null. But what else?