论文标题
关于四分之一循环图及其极端空格的无效
On the nullities of quartic circulant graphs and their extremal null spaces
论文作者
论文摘要
循环图是一个简单的图形，其邻接矩阵可以以循环矩阵的形式表示，而螺母图被认为是一个图形，其无空空间由单个完整矢量跨越。在Damnjanović[arxiv：2212.03026，2022]的先前研究中，已经确定了存在$ d $ d $的循环螺母$ n $的所有对$（n，d）$的完整集。在上述结果的激励下，我们将重点放在四分之一的循环图上，并得出了计算其无效的明确公式。此外，我们实施了上述公式，以获取一种检查特定四分之一循环图的奇异性的方法，并找到用于测试该图是否是螺母图的简明标准。随后，我们计算了固定订单$ n $的四分之一循环图的最低和最大无效，对于每个可行的订单$ n \ ge 5 $。最后，我们确定所有达到这些无效的图表，然后提供其所有相应的极端空格的完整表征。
A circulant graph is a simple graph whose adjacency matrix can be represented in the form of a circulant matrix, while a nut graph is considered to be a graph whose null space is spanned by a single full vector. In a previous study by Damnjanović [arXiv:2212.03026, 2022], the complete set of all the pairs $(n, d)$ for which there exists a $d$-regular circulant nut graph of order $n$ has been determined. Motivated by the said results, we put our focus on the quartic circulant graphs and derive an explicit formula for computing their nullities. Furthermore, we implement the aforementioned formula in order to obtain a method for inspecting the singularity of a particular quartic circulant graph and find the concise criteria to be used for testing whether such a graph is a nut graph. Subsequently, we compute the minimum and maximum nullity that a quartic circulant graph of a fixed order $n$ can attain, for each viable order $n \ge 5$. Finally, we determine all the graphs attaining these nullities and then provide a full characterization of all of their corresponding extremal null spaces.
