Nontrivial nuciferous graphs exist

TL;DR

This paper disproves a conjecture that only the complete graph K_2 is nuciferous, showing that nontrivial nuciferous graphs do exist and proposing that infinitely many such Cayley graphs may exist.

Contribution

The authors disprove the existing conjecture about nuciferous graphs and introduce the possibility of infinitely many nuciferous Cayley graphs.

Findings

Nuciferous graphs other than K_2 exist.

The conjecture that only K_2 is nuciferous is false.

There may be infinitely many nuciferous Cayley graphs.

Abstract

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.