Symmetric graphs of valency 4 having a quasi-semiregular automorphism

Highlights：

• The method in this paper is completely different from the method in Feng et al. (2019). In fact, the former paper in 2019 mainly solved the case when a stabilizer has a bound, and this paper solve the case when a stabilizer has no bound.

• The method in this paper combines group theory and graph theory, and in particular, group theory is widely used, including the well-known classification of non-abelian simple groups.

• The research on the topic of this paper began in 2013, and this was proposed by Kutnar, Malnič, Martínez and Marušič as a kind of symmetry of graphs. After 2013, nearly no much work has been done except the above mentioned work in 2019.

摘要

•The paper completely finish the classification of symmetric graphs of valency 4 having a quasi-semiregular automorphism, while its partial work for automorphism group being 2-arc-transitive or solvable was done in a former paper in Feng et al. (2019).•The method in this paper is completely different from the method in Feng et al. (2019). In fact, the former paper in 2019 mainly solved the case when a stabilizer has a bound, and this paper solve the case when a stabilizer has no bound.•The method in this paper combines group theory and graph theory, and in particular, group theory is widely used, including the well-known classification of non-abelian simple groups.•The research on the topic of this paper began in 2013, and this was proposed by Kutnar, Malnič, Martínez and Marušič as a kind of symmetry of graphs. After 2013, nearly no much work has been done except the above mentioned work in 2019.