Nuciferous Graph

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 (Ghorbani 2016).

The path graph  has adjacency matrix (and adjacency matrix inverse) given by

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.

REFERENCES

Ghorbani, E. "Nontrivial Nuciferous Graphs Exist." 18 Mar 2016. http://arxiv.org/pdf/1603.03741.pdf.

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.

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.