Research For The Matching Energies Of Some Typical Classes Of Graphs With Fixed Parameters

The matching energy of a graph is one of the important members in the family of graph energies.In 2012,Gutman and Wagner defined the matching energy of a graph G as(?),where m(G,k)denotes the number of k-matchings in G.In the literature,the Hosoya index of a graph G is defined to be Z(G)=?k?0 m(G,k).It can be observed that both the matching energy and Hosoya index are closely related to the number of matchings.The matching theory of graphs is a classical topic in graph theory,which also has profound theoretical value in the research on the matching energy and Hosoya index.Furthermore,the matching energy of graph is highly related to the energy of graph,and both of them have important applications in studying the energy of molecular(graph).Therefore,the research on the matching energy of graphs has important theoretical significance and broad application prospect.In this thesis,we use gn,t to denote the set of connected graphs with n vertices and t blocks,use Bgn,t to denote the set of connected bipartite graphs with n vertices and t blocks,use BCn,m to denote the set of bipartite cactus graphs with n vertices and m edges,use Bn,t to denote the set of book graphs with n vertices and t induced cycles,and use BBn,t to denote the set of bipartite book graphs with n vertices and t induced cycles.This thesis presents a systematic study for the matching energies of the graph classes gn,t,Bgn,t,BCn,m,Bn,t,and BBn,t.We respectively describe the graphs with the maximum matching energy in gn,t and Bgn,t and the graphs with the minimum matching energy in BCn,m,Bn,t and BBn,t.