论文标题
与有限组关联的联合订单图
The co-prime order graph associated with a finite group
论文作者
论文摘要
令$ g $为有限的组。 $ g $的联合订单图是其顶点集为$ g $的图形，如果gcd $（o（x），o（y（y））$是$ 1 $或prime，则两个不同的顶点$ x，y $相邻，其中$ o（x）$和$ o（y（y）$是$ o（x）$和$ o（y）$的订单分别为$ $ x $和$ y $。在本文中，我们表征了所有有限组的共同订单图，并对所有有限组的共同订单图为平面的所有有限组。此外，我们计算了循环群，二面体组和广义季节群的联合序列图的顶点连接性，该基团回答了Banerjee（2019）的问题。最后，我们证明，对于固定的积极整数$ k $，有限的许多有限群体的联合订单图具有（不可定位的属$ k $）。作为应用程序，我们对所有有限组的共同订单图具有（不可定位的属）分类。
Let $G$ be a finite group. The co-prime order graph of $G$ is the graph whose vertex set is $G$, and two distinct vertices $x,y$ are adjacent if gcd$(o(x),o(y))$ is either $1$ or a prime, where $o(x)$ and $o(y)$ are the orders of $x$ and $y$, respectively. In this paper, we characterize all finite groups whose co-prime order graphs are complete and classify all finite groups whose co-prime order graphs are planar. Also, we compute the vertex-connectivity of the co-prime order graph of a cyclic group, a dihedral group and a generalized quaternion group, which answers a question by Banerjee (2019). Finally, we prove that, for a fixed positive integer $k$, there are finitely many finite groups whose co-prime order graphs have (non)orientable genus $k$. As applications, we classify all finite groups whose co-prime order graphs have (non)orientable genus one and two.
