【作者】 曹磊；

【导师】 侯耀平；

【作者基本信息】 湖南师范大学 ， 基础数学， 2005， 硕士

【摘要】 设G是n阶简单图。显然,G恰有一个主特征值当且仅当G为正则图。而刻划恰有k（k≥2）个主特征值是Cvetokvi（?）提出的一个长期未解决的问题。近来A.Dress和Gutman从图中路数目的计算,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λ,使A（G）d（G）=λd（G）。其中d（G）=（d（v₁）,…,d（v_n））^T为G的度序列。从特征值上来看,一个非正则图G是λ-调和图当且仅当G恰以λ和0为主特征值。具有较少圈的连通的调和图: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）:连通的五圈调和图恰有62个。其中,连通非正则的五圈3-调和图恰有56个;连通正则的五圈3-调和图恰有5个;连通五圈4-调和图恰有1个。（2）:连通的六圈调和图恰有77个。其中,连通非正则的六圈3-调和图恰有55个;连通正则的六圈3-调和图恰有19个;连通非正则的六圈4-调和图恰有2个;连通正则的六圈4-调和图恰有1个。更多还原

【Abstract】 Let G be a simple graph of order n. clearly, G has exactly one main eigenvalue if and if G is regular. It is a long-standing problem of D.Cvetkovic to characterize graphs with exactly k(k > 2) main eigenvalues. Recently, A.Dress and Gutman gave an concept of harmonic graphs: The graph G is said to be harmonic if there exists a constant A, such that the equality A(G)d(G)=Ad(G) holds. One non-regular graph G isA-harmonic if only if G has exactly main eigenvalues A and zero. Earlier all harmonic trees were determined and all unicyclic , acyclic, bicyclic, tricyclic and tetracyclic harmonic graphs were characterized.In my paper, we go a step further and find all pentacyclic and hexacyclic harmonic graphs: (1) : There are exactly 62 connected pentacyclic harmonic graphs, where there are exactly 56 non-regular and 5 regular connected pentacyclic 3-harmonic graphs and there exist exactly 1 non-regular connected pentacyclic 4-harmonic graphs. (2) : there are exactly 77 connected hexacyclic harmonic graphs, where there are exactly 55 non-regular and 19 regular connected hexacyclic 3-harmonic graphs and there exist exactly 2 non-regular and 1 regular connected hexacyclic 4-harmonic graphs.更多还原