Infinite Families Of One-regular Cayley Graphs With Small Valency

The symmetry of graphs has been being a very hot issue problem in studying groups and graphs. For the graphs with the high symmetry, the one-regular graph and mainly with small valencies is an important researching object. Being rare and difficult to consruct, it attracts more and more scholars in this field. In this thesis, we mainly research the construction of some infinite families of one-regular Cayley graphs with valency three and four on the alternating groups.In Chapter one, we simply introduce development and current situation of one-regular graph and backgroud of one-regular graph.In Chapter two, we give some related definitions and prepositions.In Chapter three, we mainly construct the infinite families of one-regular Cayley graphs with valency three and four on the alternating group. In §3.1, we constructe six infinite families of one-regular Cayley graphs, such that for all positive integral n (n ≥ 10), we can at least construct an infinite families of one-regular graphs on the alternating group A_n with valency three. Later, we construct an infinite families of one-regular graphs on the alternating group A_n with valency three in another method.In §3.2, we given two full conditions of one-regular Cayley graphs with valency four and vertex stabilizer are Z₂ × Z₂ on the alternating group. As a appliance, we constructe four infinite families of one-regular Cayley grphs with valency four and vertex stabilizer are Z₂ × Z₂.In §3.3, we given two related theory of construct a one-regular Cayley graphs with valency four and vertex stabilizer are Z4 on the alternating group (among them, 3.2.8 is a full condition of one-regular graph). As a appliance, we constructe five infinite families of one-regular Cayley grphs with valency four and vertex stabilizer...