【作者】 陈爱莲；

【导师】 常安；

【作者基本信息】 福州大学 ， 应用数学， 2003， 硕士

【摘要】 设是个顶点的有限简单连通图，的邻接矩阵记为，的特征多项式记为，的特征值又称为的特征值。由于是非负对称不可约矩阵，故的最大特征值为单根，而且其所有特征值均为实数，从而按其大小顺序排列为：。的能量定义为。本文主要讨论单圈图类的排序问题。本文分为两大部分：第一部分讨论了单圈图类按其最大特征根从大到小进行排序的问题。设是由的一个顶点加上条悬挂边所得到的图，则洪渊已经证明了：是该序的第一个图。本文对此问题进一步讨论，确定了该序的前六个图中的另外五个图。第二部分讨论了单圈图类按其能量的排序问题。设是所有阶单圈图组成的集合，是由的一个顶点加上条悬挂边所得到的图，是由的一个顶点加上条悬挂边，另一个顶点加上一条悬挂边所得到的图。侯耀平已经证明了：有成立，且等号成立当且仅当。在此基础上，本文对此问题进一步讨论，得出下面两个结论：（1） 当时，，有且等号成立当且仅当。（2）当时，，有且等号成立当且仅当更多还原

【Abstract】 Let c be a simple connected graph with vertices. We denoted by the adjacency matrix of . All of the eigenvalue of are real, since is real and symmetric. We denote the eigenvalues of in decreasing order byand call then the eigenvalues of. The energy ofis defined as .In this thesis we discuss the problem of the ordering of unicyclic graphs. In terms of their largest eigenvalues and energy. this thesis are divided into two charpters: In the first charpter, we investigate the problem of the ordering of unicyclic graphs by their largest eigenvalues. Let be the graph obtained from the graph by adding pendant edges to one of its vertices. Hong Yuan has proved that in all unicyclic graphs, the largest eigenvalue of the graph is the greatest. On this basis, we discussed the problem further and found the other first five graphs in this order.In the second charpter, we discussed the ordering of the unicyclic graphs by their energy. Let be the graph obtained from cycle by adding pendant edges to one of its vertices, and be the graph obtained from cycle by adding pendant edges and a pendant edge to two vertices of respectively. Yaoping Hou has proved that if is a unicyclic graph with vertices and , then . On this basis, we discussed the problem further and deduced two results as follows:(1) If ，, then , with equality if andonly if 。(2) If ，，then, with equality if and only if更多还原