On The Research Of The Energy Of Trees

The energy of a graph is defined as the sum of the absolute values of its all eigenvalues.With the wider use of it in chemical realm in recent years,the research concerning about this aspect have mushroomed like an endless stream.This paper which stands on the basis of previous results,and which further research on the issue of the energy of trees,mainly includes:·(ⅰ)It introducts the paper's research background and significance,including the development of a representative at home and abroad regarding this aspect.Based on this research background and profound discussion on the status quo,it fully shows the main work's necessity and innovation.·(ⅱ)This paper mainly established on the following articles.Gutman[Acyclic systems with extremal Hückelπ-electron energy,Theoret.Chim.Acta 45(1977)79-87]characterized the trees with n vertices having the minimal,the second-minimal,the third-minimal,the fourth-minimal,the maximal,and the second-maximal energies.W.Yan,L.Ye[Appl. Math.Lett.18(2005)1046-1052],B.Zhou,F.Li[Journal of Mathematical Chemistry, 39(2006)465-473]used mathematical induction to determine the trees with the minimal and the second-minimal energies with a given diameter.F.Zhang,H.Li[On acyclic conjugated molecules with minimal energies,Discrete Appl.Math.92(1999)71-84]have determined trees with minimal,second-minimal and third-minimal energies in the class of trees with a perfect matching on n vertices.In this chapter,as an extension of the first article,we determine the trees with n vertices having the fifth-minimal,the sixth-minimal, the seventh-minimal,and the third-maximal energies;also,we determine the trees with n vertices having extremal Hosoyawe index;meanwhile,we determine the trees with a given diameter having the minimal energy by the graphic operation;as the main conclusions of the above study,we completely solve a conjecture proposed by B.Zhou,F.Li[Journal of Mathematical Chemistry,39(2006)465-473];as a consequence of the last article,using the quasiordering relation,we discuss the trees in the class of trees with a perfect matching on n vertices having the fourth-,fifth- and sixth-minimal energies.