Nontrivial nuciferous graphs exist
  A nuciferous graph is a simple graph with a non-singular $0$-$1$ adjacency matrix $A$ such that all the diagonal entries of $A^{-1}$ are zero and all the off-diagonal entries of $A^{-1}$ are non-zero. Sciriha et al. conjectured that except $K_2$, no nuciferous graph exists. We disprove this conjecture. Moreover, we conjecture that there infinitely many nuciferous Cayley graphs. 
Ebrahim Ghorbani
math.CO