论文标题
两个循环螺母图的家族
Two families of circulant nut graphs
论文作者
论文摘要
循环螺母图是一个非平凡的简单图，其邻接矩阵是无效的循环矩阵，因此其非零空空间向量没有零元素。 Bašić等人最初启动了循环螺母图的研究。 [艺术离散应用。数学。 5（2）（2021）＃p2.01]，其中对存在所有可能的对（n，d）$存在的猜想是在其中存在$ d $ d $的循环螺母$ n $。后来，Damnjanović和Stevanović证明了这一点[线性代数应用。 633（2022）127-151]，对于每个奇数$ t \ ge 3 $，使得$ t \ not \ equiv_ {10} 1 $和$ t \ not \ equiv_ {18} 15 $，$ 4T $ - 4T $ - 台词$ n $带有生成器集的订单$ n $，生成器集$ \ \ \ \ \ {1，2，2，3，3，\ ldos，2，2，\ ldos， \ {t \}）$也必须是每个$ n \ ge 4t + 4 $的螺母图。在本文中，我们通过构建两个循环螺母图系列来扩展这些结果。第一个家庭包括$ 4T $的订单$ n $循环图，该图与发电机集合$ \ {1，2，2，\ ldots，t-1 \} \ cup \ left \ left \ left \ {\ frac {n} {n} {4} {4} {4} {4}，\ frac {n} \ left \ {\ frac {n} {2} - （t -1），\ ldots，\ frac {n} {2} {2} - 2，\ frac {n} {2} {2} - 1 \ right \} $，对于每个奇数$ t \ in \ in \ in \ in \ n \ nn} $ n n} $ n} $和$ n \ ge 4 $ $ n $ 4t。第二个家庭由$ 4T $的$ N $循环图组成，该图$ n $对应于发电机集$ \ {1，2，\ ldots，t-1 \} \ cup \ left \ left \ left \ {\ frac {n+2} {4} {4} {4} {4} {4}，\ frac {n+6} \ left \ {\ frac {n} {2} - （t -1），\ ldots，\ frac {n} {2} {2} - 2，\ frac {n} {2} {2} -1 \ right \} $ 2 $。我们证明，属于这些家族的所有图形确实都是螺母图，从而完全解决了$ 4T $的循环循环螺母图订单订单订购问题问题，只要$ t $奇怪，并且可以部分解决此问题，即使$ t $的值也可以解决此问题。
A circulant nut graph is a non-trivial simple graph whose adjacency matrix is a circulant matrix of nullity one such that its non-zero null space vectors have no zero elements. The study of circulant nut graphs was originally initiated by Bašić et al. [Art Discrete Appl. Math. 5(2) (2021) #P2.01], where a conjecture was made regarding the existence of all the possible pairs $(n, d)$ for which there exists a $d$-regular circulant nut graph of order $n$. Later on, it was proved by Damnjanović and Stevanović [Linear Algebra Appl. 633 (2022) 127-151] that for each odd $t \ge 3$ such that $t\not\equiv_{10}1$ and $t\not\equiv_{18}15$, the $4t$-regular circulant graph of order $n$ with the generator set $\{ 1, 2, 3, \ldots, 2t+1 \} \setminus \{t\})$ must necessarily be a nut graph for each even $n \ge 4t + 4$. In this paper, we extend these results by constructing two families of circulant nut graphs. The first family comprises the $4t$-regular circulant graphs of order $n$ which correspond to the generator sets $\{1, 2, \ldots, t-1\} \cup \left\{\frac{n}{4}, \frac{n}{4} + 1 \right\} \cup \left\{\frac{n}{2} - (t-1), \ldots, \frac{n}{2} - 2, \frac{n}{2} - 1 \right\}$, for each odd $t \in \mathbb{N}$ and $n \ge 4t + 4$ divisible by four. The second family consists of the $4t$-regular circulant graphs of order $n$ which correspond to the generator sets $\{1, 2, \ldots, t-1\} \cup \left\{\frac{n+2}{4}, \frac{n+6}{4} \right\} \cup \left\{\frac{n}{2} - (t-1), \ldots, \frac{n}{2} - 2, \frac{n}{2}-1 \right\}$, for each $t \in \mathbb{N}$ and $n \ge 4t + 6$ such that $n \equiv_{4} 2$. We prove that all of the graphs which belong to these families are indeed nut graphs, thereby fully resolving the $4t$-regular circulant nut graph order-degree existence problem whenever $t$ is odd and partially solving this problem for even values of $t$ as well.
