The Matching Energy Of Graphs

We only consider finite, undirected, simple graphs throughout this thesis. In 2012, I.Gutman and S.Wagner proposed the concept of matching energy?ME? and pointed out that the chemical applications of matching energy go back to the 1970 s. Let G be a simple graph of order n and ?₁,?₂,…,?_n be the zeros of its matching polynomial. The matching energy of a graph G is defined to be the sum of the absolute value of ?i?i=1,2,…,n?.In this thesis, we mainly study extremal t-apex trees of order n with respect to matching energy and the minimal matching energy of ?n, m?- graphs with a given matching number. In the first chapter we briefly recall the bas ic concepts in graph theory, the definit ion of the matching energy of a graph and the related research background, present some known results. In the second chapter, we firstly introduce some important lemmas of matching energy. Then, the upper and lower bounds on matching energy are determined for t-apex tree? t?1? of order n, we characterize the corresponding extremal graphs. Moreover, as an application, the extremal cacti of order n and with s cycles are completely characterized at which the minimal matching energy is attained. In the third chapter, we consider the minimal matching energy of ?n, m?- graphs ?n?m?2n-4? and the minimal matching energy of ?n, m?- graphs ?m=n-1+t, 1?t??-1? with a given matching number ?. Moreover, we characterize the corresponding extremal graphs from the above two sets which reach the minimal matching energy, respectively. In the fourth chapter, the results obtained in this thesis are summarized and some related open problems are listed.