度至多为11的正则螺母图的存在性
Existence of regular nut graphs for degree at most 11
摘要：
坚果图是具有一维核和相应的不含零元素的特征向量的奇异图。确定$d$ -正则螺母图存在的阶数$n$的问题是Gauci，Pisanski和Sciriha最近提出的.这些命令是已知的$d \leq 4$。在这里，我们解决了所有剩余的情况下$d\leq 11$的问题，并确定了完整的列表的所有$d$ -定期螺母图的顺序$n$的小值$d$和$n$。通过计算机搜索确定了小正则螺母图的存在性或不存在性。主要的工具是一个构造，产生任何$d$ -定期螺母图的顺序$n$，另一个$d$ -定期螺母图的顺序$n + 2d$。如果我们给出足够数量的连续阶的$d$ -正则螺母图，称为种子图，这种构造可以以这样一种方式应用，即建立所有高阶的$d$ -正则螺母图的存在性。对于偶数$d$，阶数$n$确实是连续的，而对于奇数$d$，阶数$n$是连续的偶数。进一步，得到了点传递螺母图的阶和度的组合的必要条件。
英文摘要：
A nut graph is a singular graph with one-dimensional kernel and corresponding eigenverctor with no zero elements. The problem of determining the orders $n$ for which $d$-regular nut graphs exist was recently posed by Gauci, Pisanski and Sciriha. These orders are known for $d \leq 4$. Here we solve the problem for all remaining cases $d\leq 11$ and determine the complete lists of all $d$-regular nut graphs of order $n$ for small values of $d$ and $n$. The existence or non-existence of small regular nut graphs is determined by a computer search. The main tool is a construction that produces, for any $d$-regular nut graph of order $n$, another $d$-regular nut graph of order $n + 2d$. If we are given a sufficient number of $d$-regular nut graphs of consecutive orders, called seed graphs, this construction may be applied in such a way that the existence of all $d$-regular nut graphs of higher orders is established. For even $d$ the orders $n$ are indeed consecutive, while for odd $d$ the orders $n$ are consecutive even numbers. Furthermore, necessary conditions for combinations of order and degree for vertex-transitive nut graphs are derived.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
