五圈及其六圈调和图.pdf

湖南师范大学硕士学位论文五圈及六圈调和图姓名:曹磊申请学位级别:硕士专业:基础数学指导教师:侯耀平 20050301 ,(k 22),提出了调和图的概念:一个图G称为是调和的,如果存在一个数A,使 A(V)d(V)=州(G).其中d(C)=(酬u1),?,d(%))·,:如调和树,单圈图,双圈图,三圈图和四圈图都以完全确定. 本论文研究并确定了所有的五圈调和图和六圈调和图: (1):,连通非正则的五圈3一调和图恰有 56个;连通正则的五圈3一调和图恰有5个;连通五圈4一调和图恰有 1个.(2):,连通非正则的六圈3一调和图恰有55个;连通正则的六圈3_调和图恰有19个;连通菲正则的六圈4一调和图恰有2个;. 关键词:调和图; 图谱; 主特征值; 五圈图; 六圈图湖南师范大学硕士学位论文·II Abstract Let G be a simple graph oforder ,G has exactly one main eigenvalue ifand ifG is along-standing 1)roblem to characterize graphs with exactlyk(k≥2)main ,A,Dress and Gutman gave an concept ofharmonic graphs:The graph G issaid tobe harmonic ifthereexists aconstant A,such thattheequalityA(G)d(G)=Ad(G) non—regular graph G isA—harmonic ifonly ifGhasexactly main eigenvalues Aand allharmonic trees were determined and all unicyclic,acyclic,bicyclic,tricyclic and tetracyclic harmonic graphs were characterize(1. Inmy go astep further and findallpentacyclic and hexacyclic harmonic graphs:(1):There are exactly 62connected pentaeyclic hmmonic graphs,where there are exactly 56non-regular and 5regular connected pen— tacyclic 3-harmonic graphs and there exist exactly 1non—regular connected pentacyclic 4-harmonic graphs.(2):there are exactly 77connected hexacyclic harmonic graphs,where there are exactly 55lion—regular and 19regular con— nected hexacyelic 3-harmonic graphs and there existexactly 2non—regular and 1regular connected hexacyclic 4-harmonic graphs. Keywords:Harmonic graphs;Spectra ofgraphs;Main eigenvalues;Pentacyclic graphs;Hexacyclic graphs. 湖南师范大学硕士学位论文第一章绪论代数图论是目前国际上一个非常活跃的图论分支,『10,1l1,,11 , 由D,[31提出主特征值的概念以来,,相互影响,使之成