零空间为满向量张成的奇异符号图：符号螺母图
On singular signed graphs with nullspace spanned by a full vector: Signed nut graphs
摘要：
一个有符号图的边权值取自集合$\{+1,-1\}$，如果它在符号切换操作下等价于一个无符号图，则称它是符号平衡的;否则称它是符号不平衡的。坚果图有一个一维核，对应的特征向量是满的。在本文中，我们推广的概念，螺母图签署图。最近确定了度高达$11$的无符号正则螺母图存在的顺序。通过扩展的定义，有符号的螺母图，我们发现所有的对$(ρ, n)$的$ρ$ -正则螺母图（符号平衡或符号不平衡）的顺序$n$存在与$ρ\le 11$。我们设计了一个基于较小的“种子”图的符号螺母图的构造，给出了符号平衡和符号不平衡的$ρ$ -正则螺母图的无穷级数.所有订单的一个完整的符号不平衡螺母图存在的特点，他们有潜在的图形$K_n$与$n \equiv 1 \pmod 4$。所有的订单，其中一个经常的符号不平衡螺母图与$ρ= n - 2$存在的特点，他们有一个潜在的鸡尾酒会图$\mathrm{CP}(n)$，甚至$n \geq 8$。
英文摘要：
A signed graph has edge weights drawn from the set $\{+1,-1\}$, and is termed sign-balanced if it is equivalent to an unsigned graph under the operation of sign switching; otherwise it is called sign-unbalanced. A nut graph has a one dimensional kernel with a corresponding eigenvector that is full. In this paper we generalise the notion of nut graphs to signed graphs. Orders for which unsigned regular nut graphs exist were determined recently for the degrees up to $11$. By extending the definition to signed nut graphs, we find all pairs $(ρ, n)$ for which a $ρ$-regular nut graph (sign-balanced or sign-unbalanced) of order $n$ exists with $ρ\le 11$. We devise a construction for signed nut graphs based on a smaller `seed' graph, giving infinite series of both sign-balanced and sign-unbalanced $ρ$-regular nut graphs. All orders for which a complete sign-unbalanced nut graph exists are characterised; they have underlying graph $K_n$ with $n \equiv 1 \pmod 4$. All orders for which a regular sign-unbalanced nut graph with $ρ= n - 2$ exists are also characterised; they have an underlying cocktail-party graph $\mathrm{CP}(n)$ with even $n \geq 8$.
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