Title: Complete resolution of the circulant nut graph order-degree existence problem Show Chinese title

Title: 循环坚果图阶度存在性问题的完全解决

Abstract: 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 Ba\v{s}i\'c et al. and the first major results were obtained by Damnjanovi\'c and Stevanovi\'c, 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, Damnjanovi\'c 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 $n$. Show Chinese abstract

Abstract: 一个循环坚果图是一个非平凡的简单图，其邻接矩阵是一个循环矩阵，且其零空间由一个没有零元素的单个向量张成。 关于这些图，阶数-度数存在性问题可以被看作是确定所有可能的对$(n, d)$的数学问题，其中存在一个阶数为$n$的$d$-正则循环坚果图。 这个问题由 Bašić 等人提出。 并且最早的主要结果是由Damnjanović和Stevanović获得的，他们证明了对于每个奇数$t \ge 3$，使得$t\not\equiv_{10}1$和$t\not\equiv_{18}15$，存在一个$4t$-正则循环营养图，阶数为$n$，对于每个偶数$n \ge 4t + 4$。 之后，Damnjanović通过证明当$t$为奇数，$n$为偶数，并且$n \ge 4t + 4$成立时，或者当$t$为偶数，$n$满足$n \equiv_4 2$，并且$n \ge 4t + 6$成立时，必然存在一个$4t$-正则循环坚果图，其阶数为$n$。 在本文中，我们通过完全解决循环坚果图阶数-度数存在性问题，扩展了上述结果。 换句话说，我们完全确定所有可能的对$(n, d)$，其中存在阶数为$n$的$d$-正则循环坚果图。