Let
be a simple graph with nonsingular(0,1)adjacency matrix . If all the diagonal entries of the
matrix inverse are zero and all the off-diagonal entries of are nonzero, then is called a nuciferous graph (Sciriha et al. 2013,
Sciriha 2013, Ghorbani 2016).

which is therefore nuciferous. Initially, this was the only example known, and in fact, no others exist on 10 or fewer nodes (E. Weisstein, Mar. 18, 2016).
As a result, it was conjectured by Sciriha et al. (2013) that no others exist.

This conjecture was disproved by Ghorbani (2016) who found 21 Cayley
graphs examples on 24, 28, and 30 nodes.

Fowler, P. W.; Pickup, B. T.; Todorova, T. Z.; de los Reyes, R.; and Sciriha, I. "Omni-Conducting Fullerenes." Chem.
Phys. Lett., 568-569, 33-35, 2013.Ghorbani, E. "Nontrivial
Nuciferous Graphs Exist." 18 Mar 2016. http://arxiv.org/pdf/1603.03741.pdf.Sciriha,
I. "Molecular Graphs with Analogous Conducting Connections." The 4th
Biennial Canadian Discrete and Algorithmic Mathematics Conference (CanaDAM).
St. John's, Newfoundland: Memorial University of Newfoundland, 2013.Sciriha,
I.; Debono, M.; Borg, M.; Fowler, P.; and Pickup, B. T. "Interlacing-Extremal
Graphs." Ars Math. Contemp.6, 261-278, 2013.