Nuciferous Graph

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

The path graph P_2=K_2 has adjacency matrix (and adjacency matrix inverse) given by

 [1 0; 0 1],

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.

See also

(0,1)-Matrix, Adjacency Matrix, Cayley Graph, Matrix Inverse, Nonsingular Matrix

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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., 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.

Cite this as:

Weisstein, Eric W. "Nuciferous Graph." From MathWorld--A Wolfram Web Resource.

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