Research On The Spectral And Structure Parameters Of Graphs And Its Applications

Graph spectral theory is of great importance in algebraic graph theory and combinatorial matrix theory.It mainly uses the spectral parameters described by the matrices associated with graphs to characterize the structural properties of the graphs,and studies the intrinsic relationships between the spectral parameters of graphs and their granh structures.In this thesis,using the spectral parameters,we study the spectral parameters including Laplacian spectra,normalized Laplacian spectra,and structure parameters of graphs and their related problems.This thesis consists of five chapters.In the first chapter,we give some definitions and notations which are needed later.After that,we introduce the background and significance of the research,including the development at home and abroad.Based on this research background and profound discussion,by using deep-going analysis,it fully shows the main work's necessity and innovation.Then some lemmas will be introduced that used later.In the second chapter,we will do research on linear phenylenes,denoted by Ln(see fig1)and Ln'(see fig2)respectively which is a molecular graph obtained by attaching 4-membered cycles to the terminal hexagons.The block matrix of the linear phenylenes is diagonalizable and the decomposition of Laplacian characteristic polynomial is given.By applying the relationship between the roots and coefficients of the characteristic polynomial of the above matrix,explicit closed formula of the Kirchhoff index and the number of spanning trees are derived.Furthermore,we find the Kirchhoff index is approximately to half of its Wiener index.In the third chapter,we will do research on linear octagonal-polyomino diagonal chain,denoted by En,which is a molecular graph obtained by attaching 4-membered cycles to the terminal octagon among which there are two edges(see fig 3).The block matrix of En is diagonalizable and the decomposition of Laplacian characteristic polynomial is given.By applying the relationship between the roots and coefficients of the characteristic polynomial of the above matrix,explicit closed formula of the Kirchhoff index and the number of spanning trees are derived.Furthermore,we find the Kirchhoff index is approximately to 7/16 of its Wiener index.In the fourth chapter,we will do research on Mobius phenylene chain,denot-ed by HMn6,4.The Mobius phenylene chain HMn6,4 is the graph obtained from the Ln by identifying the opposite lateral edges in reversed way.Using the decomposi-tion theorem of the normalized Laplacian characteristic polynomial,we study the normalized Laplacian spectrum of HMn6,4,which consists of the eigenvalues of two symmetric matrices.By investigating the relationship between the roots and coef-ficients of the characteristic polynomial of the two matrices above,explicit closed formula of the degree-Kirchhoff index and the number of spanning trees are derived.Furthermore,we find the degree-Kirchhoff index is approximately to one-third of its Gutman index.In the end,we summarize the main contents and put forward some problems for further research.