Researches On The Matching Number Of Graphs

The Hosoya index of a graph G is defined to be the total number of matchings in G,which was proposed by Hosoya in 1971.In 2012,Gutman and Wagner first proposed the concept of matching energy of a graph G,which is defined as the sum of absolute values of zeros of the matching polynomial of a graph.The Hosoya index and the matching energy are two invariants of a graph.Some results about the description of the matching energy have been obtained by applying quasi-order of matching numbers.In this thesis,we mainly discuss the matching number of the complements of bicyclic graphs and trees.In the second chapter of this thesis,we show that the maximal graphs in the com-plements of the first type and the second type bicyclic graphs are (?) and(?),respectively.Then by comparing the two maximal graphs,the unique maximal graph in the complements of bicyclic graphs can be determined as (?).Similarly,we show that the minimal graphs in the complements of the first type and the second type bicyclic graphs are (?) and (?),respectively.Then by comparing the two minimal graphs,the unique minimal graph in the complements of bicyclic graphs can be determined as (?).In the third chapter of this thesis,we show that (?),(?) and (?) has the third-largest,second-smallest and third-smallest matching numbers among all complements of trees with n vertices.