A nut graph is a graph on
vertices with adjacency matrix such that has matrix rank 1 and contains
no 0 element (Sciriha 1998, 2008; Sciriha and Gutman, 1998; and Sciriha and Fowler
2008). The numbers of nut graphs on , 2, ... vertices are 0, 0, 0, 0, 0, 0, 3, 13, 560, 12551,
2060490, 208147869, 96477266994, ... (House of Graphs).
Coolsaet, K.; Fowler, P. W.; And Goedgebeur, J. "Generation and Properties of Nut Graphs." MATCH Commun. Math. Comput. Chem.80,
423-444, 2018.Damnjanovi'c, I. "Complete Resolution of the Circulant
Nut Graph Order-Degree Existence Problem." 6 Dec 2022. https://arxiv.org/abs/2212.03026.Gauci,
J. B.; Pisanski, T.; And Sciriha, I. "Existence of Regular Nut Graphs and
the Fowler Construction." 12 Nov 2019. https://arxiv.org/abs/1904.02229.Fowler,
P. W.; Gauci, J. B.; Goedgebeur, J.; Pisanski, T.; and Sciriha, I. "Existence
of Regular Nut Graphs for Degree at Most 11." Disc. Math. Graph Th.40,
533-557, 2020.House of Graphs. "Nut Graphs." https://hog.grinvin.org/meta-directory/nut.Sciriha.
I. "On the Construction of Graphs of Nullity One." Disc. Math.181,
193-211, 1998.Sciriha, I. "Coalesced and Embedded Nut Graphs in
Singular Graphs." Ars Math. Contemp.1, 20-31, 2008.Sciriha,
I. and Fowler, P. W. "On Nut and Core Singular Fullerenes." Disc.
Math.308, 267-276, 2008.Sciriha, I. and Gutman, I. "Nut
Graphs: Maximally Extending Cores." Util. Math.54, 257-272, 1998.
vertices with adjacency matrix 
such that 
has matrix rank 1 and contains
no 0 element (Sciriha 1998, 2008; Sciriha and Gutman, 1998; and Sciriha and Fowler
2008). The numbers of nut graphs on 
, 2, ... vertices are 0, 0, 0, 0, 0, 0, 3, 13, 560, 12551,
2060490, 208147869, 96477266994, ... (House of Graphs).
-antiprism
graphs are nut graphs when 
is not divisible by 3.
and vertex degree 
exists for the following values and no others:
even, 
, and 
,
and 
,
even and 
,
even, 
, and 
, or
even, 
, and 
. The complete set of circulant nut graphs on 16 or
fewer vertices are illustrated above.
