• 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. Feng et al. (2019) characterized connected G -symmetric graphs of valency 4 having a quasi-semiregular automorphism, namely, a graph automorphism fixing a unique vertex in the vertex set of the graph and keeping the lengths of all other orbits equal, when G is soluble or the vertex stabilizer in G is not a 2-group. In this paper we prove that a connected symmetric graph with valency 4 having a quasi-semiregular automorphism is a Cayley graph on a group G with respect to S , where G is abelian of odd order and S is an orbit of a group of automorphisms of the group G .
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