A Related Study Of Chip-firing And Burning Number On The Less-circle Graph

Graph theory is a branch of mathematics that deals with graphs.The dynamic behavior of the graph is a hot topic nowadays.In this paper,we mainly study the chip-firing process and burning process of the two dynamic systems on the few cycles graph.The chip-firing process on a infinite path is actually a combination game,that is,the initial state is to give a path,specify an origin and place(?)chips at the origin,we take(?)chips to move one unit to the left and right sides at the beginning of fire.In the process of each step,the same rules are followed as before.When each vertex satisfies that the number of vertex chips is less than the vertex degree,the firing stops.The burning process of the graphis a discrete-time process.When=0in the initial state,all the vertices are unburned.In each step of?1,we will choose one new unburned vertex to burn(if the vertex is available)and we will call it the source of fire.Once the vertex is burned,and it will keep burning until the end of the process.When a new vertex is burned at step,the adjacent unburned vertices of all burned vertices will become burned at step+1.The process is finished until all vertices in graphare burned.In Chapter 1,we first give some basic theoretical knowledge of the graph and the basic concepts and properties of the two dynamic behavior.Then,we introduce the significance,development and recent research status of the two dynamic behavior.Finally,we describe the main work of this paper.In Chapter 2,the positions of origin and sink are given on the cycle graph,and the configuration and total number of firing are presented when the chip-firing process is stable.In Chapter 3,we study the burning process of the line graph of spider tree with three components,and give the burning number of the line graph of spider tree with three components.In Chapter 4,we study the burning process of two basic bicyclic graphs.We discuss the burning number of?-graph and the dumbbell.In the last chapter,we give the summary and prospect of this paper.