  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.
