A Study Of Cubic Bi-Cayley Graphs Over Two Classes Of Nonabelian Groups

Algebraic graph theory is an interdisciplinary subject produced by the combination of algebra and graph theory.It mainly studies the property of graphs by means of algebraic knowledge.The graph X is a vertex transitive graph or edge transitive graph,if the full automorphism group is transitive on the vertex set or edge set of X,respectively.A graph is said to be arc transitive if X has no isolated vertices and its full automorphism group is transitive on the arc set of X.A graph is said to be a Cayley graph over a group H,if it has a regular automorphism group isomorphic to H.A graph is a bi-Cayley graph over a group H,if it has a semiregular automorphism group which is isomorphic to H with exactly two orbits.The graph called nonzero component graph where the vertex set is the collection of nonzero vector of a finite dimensional vector space and two vertices are adjacent if they share at least one basis with nonzero coefficient in their basic representation.As we know,graph automorphism group can well reflect the symmetry of graphs,and graph classification can more systematically apprehend the properties and structures of graphs.Therefore,this paper mainly studies the automorphism group of graphs and the classification of graphs.Firstly,we mainly study the cubic bi-Cayley graphs over the generalized quaternion group,give the classification of bi-Cayley over the generalized quaternion group,and determine the automorphism group of the graphs.Secondly,we mainly study the cubic bi-Cayley graphs over the group of order pq,give the classification of bi-Cayley graphs over the group of order pq,and determine the graph automorphism groups.Finally,we investigate regular subgraphs of nonzero component graphs.The property of graphs is given.In addition,we obtain three classes of regular subgraphs of nonzero component graph,determine the automorphism groups of these three classes of graphs,and prove that two of them are lexicographic products.In addition,the background,definition and conclusion of bi-Cayley graph and nonzero component graph are introduced in the first and second chapters of this paper.In Chapter 6,the main conclusions of paper are given and we introduces some problems to be solved.