Study On Some Properties Of(Unit) Bi-Cayley Graphs

Algebraic graph theory is one of the important research fields of graph theory,which mainly use algebraic methods to solve graph theory problems.Algebraic graph theory has three main branches,which are graph and linear algebra,graph and group and graph invariants.Graph and group mainly study of the relationship between graphs and group,while Cayley graphs are graphs constructed by using the elements of the group as the vertices of the graph and then specifying a law as the edges of the graph.Bi-Cayley graph(also known as semi-Cayley graph)is a generalization of Cayley graph,many famous graphs are bi-Cayley graph,such as Peterson graph,generalized Peterson graph and Hoffman-Singleton graph and so on,but these graphs cannot be constructed from Cayley graphs.In this paper,we first characterize the equivalent conditions for unitary one-matching bi-Cayley graphs over finite commutative rings and its complement and line graph to be Ramanujan,and give the spectral moments of unitary one-matching bi-Cayley graphs.Secondly,we study the spectral of quadratic unitary one-matching bi-Cayley graphs over finite commutative rings,and determine when such a graph is Ramanujan.Finally,the energy and Randi(?) energy of the division graph,Q-graph and R-graph of unitary one-matching bi-Cayley graph and unitary homogeneous bi-Cayley graph are described by using the matrix analysis method and combining the spectral graph theory knowledge.The full content consists of the three chapters as follows:In Chapter 1,some basic concepts,research background and progress of several classes(bi)Cayley graphs are introduced,and the main results of this thesis are lieted.In Chapter 2,we describe the equivalent conditions for unitary one-matching biCayley graphs over finite commutative rings and its complement and line graph to be Ramanujan,and give the spectral moments of unitary one-matching bi-Cayley graphs.We also study the spectral of quadratic unitary one-matching bi-Cayley graphs over finite commutative rings,and determine when such a graph is Ramanujan.In Chapter 3,we study the energy and Randi(?) energy of division graph,Q-graph and R-graph of unitary one-matching bi-Cayley graph and unitary homogeneous biCayley graph.