18pq阶7度对称图的分类

Classification of Heptavalent Symmetry Graphs of Order 18pq

摘要: 若一个图的自同构群在其弧集合上作用传递，则称该图为对称图。郭等人于2018年、2020年完成了阶为$18p$ 与$36pq$ 的全部七度对称图的分类。本文推广上述结果，对满足素数条件$q>p\ge 3$ 的阶为$18pq$ 的所有七度对称图进行分类。作为推论，得到两个具体图，记为${\mathcal{C}}_{990}^1$ 与${\mathcal{C}}_{990}^2$ ，以及三个具体的自同构群，即$\text{PSL}\left(2,503\right)$ 、$\text{PSL}\left(2,757\right)$ 和$\text{PGL}\left(2,757\right)$ 。

Abstract: A graph is symmetric if its automorphism group acts transitively on its arc set. Recently, Guo et al. (2018, 2020) completed the classification of all heptavalent symmetric graphs of order $18p$ and $36pq$ . We generalize these results here by classifying all heptavalent symmetric graphs of order $18pq$ for primes $p,q$ with $q>p\ge 3$ . Consequently, two concrete graphs, denoted ${\mathcal{C}}_{990}^1$ and ${\mathcal{C}}_{990}^2$ , and three concrete automorphism groups, namely $\text{PSL}\left(2,503\right)$ , $\text{PSL}\left(2,757\right)$ and $\text{PGL}\left(2,757\right)$ , are obtained.