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循环坚果图阶数-度数存在性问题的完整解决

Complete resolution of the circulant nut graph order-degree existence problem

Ivan Damnjanović

备注：
arXiv admin note: text overlap with arXiv:2210.08334
发表信息：
Ars Math. Contemp. 24 (2024) #P4.03
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AI总结：

本文完整解决了循环坚果图的阶数-度数存在性问题，完全刻画了存在 $d$-正则循环坚果图的所有可能数对 $(n, d)$，推广了 Damnjanović 等人的已有结果。

摘要：

循环坚果图是一类非平凡简单图，其邻接矩阵为循环矩阵，且其零空间由一个不含零元素的单一向量张成。关于这类图，阶数-度数存在性问题可视为如下数学问题：确定所有可能的数对 $(n, d)$，使得存在一个阶数为 $n$ 的 $d$-正则循环坚果图。该问题由 Bašić 等人提出，首个主要结果由 Damnjanović 和 Stevanović 获得，他们证明了对于每个奇数 $t \ge 3$，只要 $t\not\equiv_{10}1$ 且 $t\not\equiv_{18}15$，则对每个偶数 $n \ge 4t + 4$，都存在一个阶数为 $n$ 的 $4t$-正则循环坚果图。此后，Damnjanović 改进了这些结果，证明了当 $t$ 为奇数、$n$ 为偶数且 $n \ge 4t + 4$ 时，或者当 $t$ 为偶数、$n$ 满足 $n \equiv_4 2$ 且 $n \ge 4t + 6$ 时，必然存在一个阶数为 $n$ 的 $4t$-正则循环坚果图。本文通过完整解决循环坚果图阶数-度数存在性问题，推广了上述结果。换言之，我们完全确定了所有可能的数对 $(n, d)$，使得存在一个阶数为 $n$ 的 $d$-正则循环坚果图。

英文摘要：

A circulant nut graph is a non-trivial simple graph such that its adjacency matrix is a circulant matrix whose null space is spanned by a single vector without zero elements. Regarding these graphs, the order-degree existence problem can be thought of as the mathematical problem of determining all the possible pairs $(n, d)$ for which there exists a $d$-regular circulant nut graph of order $n$. This problem was initiated by Bašić et al. and the first major results were obtained by Damnjanović and Stevanović, who proved that for each odd $t \ge 3$ such that $t\not\equiv_{10}1$ and $t\not\equiv_{18}15$, there exists a $4t$-regular circulant nut graph of order $n$ for each even $n \ge 4t + 4$. Afterwards, Damnjanović improved these results by showing that there necessarily exists a $4t$-regular circulant nut graph of order $n$ whenever $t$ is odd, $n$ is even, and $n \ge 4t + 4$ holds, or whenever $t$ is even, $n$ is such that $n \equiv_4 2$, and $n \ge 4t + 6$ holds. In this paper, we extend the aforementioned results by completely resolving the circulant nut graph order-degree existence problem. In other words, we fully determine all the possible pairs $(n, d)$ for which there exists a $d$-regular circulant nut graph of order $n$.

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。

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