【作者】 沈小玲；

【导师】 张远平； 侯耀平；

【作者基本信息】 湖南师范大学 ， 基础数学， 2005， 硕士

【摘要】 “哪些图由它的谱确定?”的问题于半个世纪前起源于化学。1956年Günthard和Primas在一篇把图谱理论与化学中Hückel’s理论相联系的论文中提出了该问题。另一个应用来自于1966年Fisher考虑的Kac提出的一个问题:“一个人能否听到鼓声的形状?”。他用一个图模拟了鼓声的形状,那么鼓声就由该图的特征值来刻画了。Kac的问题实际上也是我们的问题。 目前对该问题特别是“哪些图由它的邻接谱或Laplacian谱确定?”的研究结果并不多。半个世纪以来,出现了一些研究正则图和特殊结构图的线图的谱刻画的文章。而对于非正则图的研究结果极为少见。本文主要研究一些特殊结构的非正则图的谱确定问题,得到了如下一些结论: （1） 图Z_n（一类特殊的似星树）既能由它的邻接谱确定,又能由它的Laplacian谱确定;进一步,最大邻接特征值小于2的图既能由它的邻接谱确定,又能由它的Laplacian谱确定。 （2） 图Z_n的不相交的并Z_n1+Z_n2+…+Z_nk能由其邻接谱确定,其中,n₁,…,n_k都是大于1的正整数;k个不相交的图Z_n的并kZ_n能由其Laplacian谱确定; （3） 似星树（包括星图）由它的Laplacian谱确定; （4） 梳图由它的Laplacian谱确定; （5） 饱和烷烃分子C_nH_2n+2的分子图由它的Laplacian谱确定; （6） W_n和其它两类似双星图虽不能由它们的邻接谱确定,但能由它们的Laplacian谱确定; （7） 恰有一个Laplacian特征值大于3的树由它的Laplacian谱确定; （8） 恰有两个Laplacian特征值大于2的树（包括双星图）由它的Laplacian谱确定;更多还原

【Abstract】 The question " which graphs are determined by their spectra? " goes back for half a century, and originates from chemistry. In 1956, Giinthard and Primas raised the question in a paper that relates the theory of graph spectra to Hucke’s theory from chemistry. Another application conies from Fisher in 1966. who considered a question of Kac: " Can one hear the shape of a drum?" He modelled the shape of the drum by a graph. Then the sound of that drum is characterized by the eigenvalues of the graph. Thus Kac’s question is essentially ours.Answering the question for adjacency or Laplacian matrices seems out of research. In the past half century, there are many results about the question for regular graphs and some special line graphs. But for non-regular graphs, there are few about it. In this paper, we prove that(1) Zn is determined by its adjacency spectrum as well as the Laplacian spectrum. Furthermore, graphs with largest adjacency eigenvalue less than 2 are determined by their adjacency spectra as well as Laplacian spectra.(2) the disjoint union of graph Z_n1, Z_n2, Z_nk (denoted by Z_n1 +Z_n2 + ... + Z_nk) is determined by its adjacency spectrum, where all the numbers n1,n2,..., n_k are positive integer greater than one: k disjoint union of Z_n is determined by its Laplacian spectrum;(3) starlike trees (including the star) are determined by their Laplacian spectra;(4) comb is determined by its Laplacian spectrum;(5) a kind of molecular graph of alkyl is determined by its Laplacian spectrum;(6) graph W_n and the other two double starlike graphs are not determined by their adjacency spectra, but they are determined by their Laplacian spectra,更多还原