Algebraic Characterization Of Fan Graphs And Its Applications

Systems engineering is a multidisciplinary,cross-cutting discipline that studies large and complex systems.Complex systems can often be abstracted as graphs,which are mathematical models used to characterize certain relationships between things.In graphs,the vertices represent objects in a complex system,and the edges between vertices represent some relationship between objects.Graph theory is the study of graphs and is divided into a number of branches according to their methods of study.The use of matrices to describe vertex-edge relations in graph theory and the application of algebraic tools to graph theory is known as algebraic graph theory,of which the Terwilliger algebra and the adjacency algebra of graphs are important studies in algebraic graph theory.The Terwilliger algebra of a graph is defined relative to a fixed basic vertex and contains adjacency algebras as subalgebras that reflect more local structural information.In this thesis,will consider the Terwilliger algebra of the fan graph.First of all,it is proved that Terwilliger algebra keep isomorphism under the action of automorphism group on graph and the automorphism group of the fan graph is given.Then,we determine the structure of irreducible modules for each of Terwilliger algebras of the fan graph.As a result,we obtain a necessary and sufficient condition for the isomorphism between the Terwilliger algebra for a fixed vertex and the centralizer algebra of stabilizer of the automorphism group of the fan graph.Considering that the continuous-time quantum walk is closely related to the adjacency algebra of the graph,this thesis works on the perfect state transfer of the fan graph with the adjacency algebra of the graph.First,the properties of different vertex pairs with perfect state transfer in the graph are introduced,and the necessary and sufficient conditions for judging strongly cospectral the vertex pairs are given.Secondly,using the characteristic polynomials of the corona graph,the spectrum of the fan graph is described,and when the order of vertex set of the fan graph is greater than 3 and is prime,the relationship between the fan graph and the characteristic polynomials of its induced subgraphs is established.Finally,a sufficient condition is given that the fan graph does not have perfect state transfer.