Researches On Matching Energy And Hosoya Index For Three Types Of Graphs

Matching energy,as an important direction in graph theory research,was proposed by Gutman and Wagner in 2012.Matching energy is defined as the sum of the absolute of the roots for the matching polynomial of a graph.In this thesis,based on the properties of the matching polynomial we mainly use the formula?(G,x)=?(G-e,x)-?(G\{u,v},x)and the recurrence formula?(Pa)?(Pb)-?(Pa-k)?(Pb+k)=?(Pa-1)?(Pb-1)-?(Pa-k-1)?(Pb+k-1)to derive the matching polynomial for three types of graphs,i.e.,heart-shaped graph,binaural graph and rabbit ear graph.Then using the method of quasi-sequence we to compare the size according to their matching numbers.In the end,we obtain the order on the matching energy of the heart-shaped graphs,binaural graphs and its complement graphs,and rabbit ear graphs and its complement graphs,and the Hosoya indexes.The whole thesis is divided into five chapters.In Chapter 1,firstly,we mainly introduce the matching energy of the graph,the Hosoya index and their background with some literature reviews;secondly,we describes the basic definitions and theorems that will be used in the next chapters.In Chapter 2,we investigate the the sorting problem for matching energy and Hosoya index for heart-shaped graphs and give their complete order.Furthermore,the graph structure of the two indexes achieving the extreme values is described.In Chapter 3,the definition of binaural graphs is introduced,and the complete order of the two indexes for the binaural graphs and its complements is given.In Chapter 4,the matching energy of rabbit ear graphs and its complement graphs and the complete order of their Hosoya index are also given.In Chapter 5,we conclude the whole thesis and point out the future work.