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个顶点上的图，其邻接矩阵 
的矩阵秩为 1 且不包含 0 元素（Sciriha 1998, 2008；Sciriha 和 Gutman，1998；以及 Sciriha 和 Fowler 2008）。顶点数为 
, 2, ... 的 Nut 图的数量为 0, 0, 0, 0, 0, 0, 3, 13, 560, 12551, 2060490, 208147869, 96477266994, ... (House of Graphs)。
-反棱柱图在 
不能被 3 整除时是 Nut 图。
且顶点度为 
时，存在以下值的循环 Nut 图，且仅存在这些值
为偶数，
，且 
,
且 
,
为偶数且 
,
为偶数，
，且 
，或
为偶数，
，且 
。顶点数不超过 16 的循环 Nut 图的完整集合如上所示。