论文标题
在单数签名的图形上，由完整矢量跨越零空间：签名的螺母图
On singular signed graphs with nullspace spanned by a full vector: Signed nut graphs
论文作者
论文摘要
签名的图具有从集合$ \ {+{+1，-1 \} $中得出的边缘权重，如果在符号切换的操作下与未签名的图等同，则称为符号平衡；否则，它被称为签名。螺母图具有一个一维内核，具有相应的特征向量已满。在本文中，我们将坚果图的概念推广到签名的图。最近确定了未签名的常规螺母图的订单，该学位最高$ 11 $。通过将定义扩展到签名的螺母图，我们找到了所有对$（ρ，n）$的$（$ρ$ regartular nut图（符号平衡或符号平衡或不平衡）的订单$ n $），均为$ρ\ le 11 $。我们根据较小的“种子”图设计了一个签名的螺母图的结构，从而提供了无限的签名平衡和签名$ρ$的螺母图。所有存在完整的标志性螺母图的订单均表征；他们的基础图$ k_n $，带有$ n \ equiv 1 \ pmod 4 $。还均表征了所有具有$ρ= n -2 $的常规标记螺母图的订单；他们的基础鸡尾酒会图$ \ mathrm {cp}（n）$，甚至$ n \ geq 8 $。
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 $n \geq 8$.
