Some Extremal Results Related To A Poset Of Trees

Graph theory mainly studies the internal structure contained in graphs,the Wiener index,the number of closed walks and the coefficients of polynomials are the important parameters in studying the structure of graphs.A tree is a connected acyclic graph.In this thesis,Using a unified approach.applying some graph shifts and recurrence relation,we study the extremal trees with the maximum(resp.minimum)Wiener index,number of closed walks,coefficients of polynomials(the adjacency polynomial,the Laplacian polynomial,the edge cover polynomial and the independence polynomial)in absolute value among the trees of order n which are neither a path nor a star.respectively.The main results are in the following:·In Chapter 1.we introduce the background of the research.According to a systematic analysis of this research background,it fully shows the main work's necessity and innovation.·In Chapter 2,we give some necessary notations,terminologies and lemmas.·In Chapter 3,for all positive integers n and l,we characterized the unique tree having the maximum(resp.minimum)Wiener index and the unique tree having the maximum(resp.minimum)number of closed walks of length l among the trees of order n which are nither a path nor a star.·In Chapter 4.for all positive integers n,we characterize the unique tree whose adjaccncy polynomial has the maxiumm(res]).minimum)coefficients in abso-lute value among the trees of order n which are neither a path nor a star.·In Chapter 5.for all positive integers n.we characterize the unique tree whose Laplacian polynomial has the maximum(resp.minimum)coefficients in abso-lutee value amongg the trees of order n which are neither a path nor a star.·In Chapter 6.for all positive integers n,wc characterize the unique tree whose independenee polynomial has the maximum(resp.minimum)coeffieicnts.the unique tree whose edge cover polynomial lias the maximm coefficients and allthe trees whose edge cover polynonial has the minimm coefficients among thetrees of order n which are neither a path nor a star.·In the last chapter,we give some further research problems along this direction.