The Linear Preserver Problems On Graphs

The preserver problem in matrix space has a long history.It mainly concerns the characterization of mappings that preserve some special properties of matrices or operators.Linear preserver problem is a class of basic and important preserver prob-lems.It has important theoretical values and wide range of applications in differential equations,control systems and quantum mechanics.In the past decades,fruitful results have been achieved on preserver problems of matrices and operators.Hershkowitz first studied the preserver problems on graph.He proved the mapping that maps the set of graphs without a circle of length longer than k into its own is bijection.Based on this,we study several kinds of preserver problems on graphs and characterize completely linear mappings that preserve certain properties of graphs.The thesis has four parts.The first chapter is the introduction,which mainly introduces the background and development status on preserver problems.We also give the symbols and definitions needed in this chapter.In chapter 2 we character-ize completely linear transformations that map Kt-free graphs to Kt-free graphs.In chapter 3 we characterize completely linear transformations that map Ck-free graphs to Ck-free graphs.In chapter 4 we characterize completely linear transformations that preserve a given independent number of all graphs.