Properities Of Bakry-(?)mery Curvature And Upper Bound Estimates Of Distances

In this paper,Bakry-Emery curvature is studied from two aspects based on several basic assumptions of locally finite discrete graphs.In the first chapter,we give some general notations and partial definitions.In the second chapter,we introduce some basic properties of the curvature function KG,x(see definition 1.3).In 2021,David Cushing,Supanat Kamtue,Shiping Liu and Norbert Peyerimhoff prove that[1]if the dimension of the eigenspace of ?min(AN)is not less than 2,then KG,x is a constant over[N,?].Our work in this chapter is to prove Theorem 2.6:If there exists N ?(0,?],such that KG,x is a constant over[N,?],then for any N1? N?N2??,the dimension of Emin(AN)is not less than the dimension of Emin(AN2).(Emin(AN)is the minimal eigenspace of the matrix AN,AN see(1.5)).We also prove a further result,namely Theorem 2.7:If there exists N ?(0,?),such that KG,x is a constant over[N,?],then the dimension of Emin(AN)is a constant over[N1,N2]if and only if v0 ? Emin(AN1).(v0 see(1.6)).In the third chapter,we give the metric pk under locally finite graphs and in-troduce rk.On this basis,we extend the discrete Bonnet Myers theorem of Shiping Liu,Florentin Munch and Norbert Peyerimhoff to the case of rk.