论文标题
完全解决循环螺母图订单的存在问题
Complete resolution of the circulant nut graph order-degree existence problem
论文作者
论文摘要
循环螺母图是一个非平凡的简单图，它的邻接矩阵是一个循环矩阵，其无元素跨越无元素的循环空间。关于这些图，可以将订单度的存在问题视为确定存在的所有可能对$（n，d）$的数学问题，其中存在$ d $的循环循环螺母图$ n $。这个问题是由Bašić等人引发的。第一个主要结果是由Damnjanović和Stevanović获得的，他们证明了每个奇数$ t \ ge 3 $ 3 $，以至于$ t \ not \ equiv_ {10} 1 $和$ t \ not \ equiv_ {18} 15 $，每个$ 4T $ ge ge ge ge for cord $ n $ n $ n $ n $ n $ n $ n \ 4 \ 4 \ 4 \ 4之后，dammjanović通过表明$ 4T $的循环循环系统$ n $时，每当$ t $奇怪，$ n $偶数，$ n \ ge 4t + 4 $持有，或者$ n $ n nesh $ n $什至什至$ n $，$ n $是如此$ n \ equiv_4 2 $ n \ e 4t $ n $ n时，在本文中，我们通过完全解决了循环螺母图阶的存在问题来扩展上述结果。换句话说，我们充分确定存在$ d $的循环螺母图$ n $的所有可能对$（n，d）$。
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 $n$.
