Bi-Cayley Graphs Over The Semidihedral Groups

Group theory and graph theory have always been mathematical problems that people pay close attention to.The combination of the two has become a new field of mathematics since the 1930s and 1940s.In recent years,with the rapid development of large-scale computer systems and interconnection networks,the research of algebraic graph theory becomes more and more important.The symmetry of graphs is an important topic in algebra graph theory.If the automorphic group of graph X transitively acts on the vertex set,then the graph X is a vertex transitive graph.If the graph X has a regular automorphism group isomorphic to H,then the graph X is a a Cayley graph over a group H.If the graph X has a semiregular automorphism group which is isomorphic to H with exactly two orbits,then the graph X is a bi-Cayley graph over a group H.In the study of Cayley graph,there is a larger project which aims at obtaining a deeper understanding of various classes of symmetric graphs.The bi-Cayley graphs are also important in the study of symmetry properties of graphs.In this paper,by using the knowledge of finite group theory,graph theory,permutation group theory and the operation of Magma software,we study the cubic bi-Cayley graph over semidihedral groups.By studying the structure of the semidihedral group and combining the analysis of bi-Cayley graphs,cubic 0-type,1-type and 2-type bi-Cayley graphs over semidihedral groups were classified.Under the isomorphism meaning,we obtain eight families vertex transitive bi-Cayley graphs and study the automorphism group of them.In addition,under the |R|=|L| conditions,we investigate several tetravalent bi-Cayley graphs over semidihedral groups,and obtain four families vertex transitive bi-Cayley graphs.