Study On Connectivity And Hamiltonian Properties Of Intersection Graphs Of Bases And Second Order Circuit Graphs Of Matroids

As an important branch of mathematics,graph theory is widely used in many fields and plays a very important role.Matroid theory,closely related to graph theory,has developed for decades,and the concept is first proposed by Whitney in 1935.Matroids is an abstraction of certain important concepts of graph theory and linear algebra,drawing on lots of symbols and representations in graph theory and linear algebra.With the steady development of matroid theory,it has been widely applied to many fields,such as combinatorial mathematics,computer science,and network theory,etc.In this thesis,we give a further study to the Hamiltonian properties and connectivity of intersection graphs of bases and the second order circuit graphs of matroids on the basis of works of Li and Zhang.Firstly,we show that the intersection graphs of bases of matroids with rank at least 3 and without coloop are Hamiltonian connected,1-vertex-fault-tolerant Hamiltonian,E₂-Hamiltonian,1-Hamiltonian connected,edge-pancyclic,and for any two edges in the intersection graphs of bases,there is Hamiltonian cycle avoiding them both.In addition,it proves that that The connectivity of the intersection graph of bases of the uniformly matroidsU_3,nis3（?）,and a lower bound on connectivity of the intersection graph of bases of matroids is given by using recursion to write the pseudo-code.About the connectivity and Hamiltonian properties of circuit graphs of matroids,many conclusions have been made.The higher the order of circuit graphs of matroids,the more complex the structure requirements of matroids,so the research of second order circuit graphs and higher order circuit graphs of matroids are yet few.Based on the previous studies,we study that the second order circuit graphs of certain representative matroids,and deduces that the second order circuit graphC₂（M（W _n））of wheel cycle matroid is uniformly Hamiltonian and the connectivity is 2n-3;the second order circuit graphs of a class of uniform matroids is complete graphs.It also proves that the second order circuit graph ofU_m-1,nis isomorphic to the second order intersection graph of bases ofU_m,n.Thus the second order intersection graph of bases ofU_m,n has the same connectivity and Hamiltonian properties as the second order circuit graph ofU_m-1,n.