which is therefore nuciferous. Initially, this was the only example known, and in fact, no others exist on 10 or fewer nodes (E. Weisstein,
pers. comm., 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.
Among graphs whose maximum vertex degree is at most 3, is the only nuciferous graph (Sciriha and Farrugia 2016,
pp. 401-402).
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. https://arxiv.org/abs/1603.03741.House
of Graphs. Nuciferous Graphs. K2.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.Sciriha,
I. and Farrugia, A. "No Chemical Graph on More Than Two Vertices Is Nuciferous."
Ars Math. Contemp.11, 397-402, 2016. https://doi.org/10.26493/1855-3974.1030.9d9.
is a simple graph with nonsingular (0,1) adjacency matrix 
such that all the diagonal entries of
the matrix inverse 
are zero and all the off-diagonal
entries of 
are nonzero (Sciriha et al. 2013, Sciriha 2013,
Ghorbani 2016).
has adjacency matrix
(and adjacency matrix inverse)
given by
![[0 1; 1 0],](/images/equations/NuciferousGraph/NumberedEquation1.svg)
is the only nuciferous graph (Sciriha and Farrugia 2016,
pp. 401-402).
