关于特殊图的Laplacian能量的研究

发布时间：2018-04-05 16:00

  本文选题：特殊图　切入点：拉普拉斯能量　出处：《大连海事大学》2017年硕士论文

【摘要】：著名的格尼斯堡七桥问题是图论问题的起源,随后图论便成为应用数学研究中的一个重要分支。图的Laplacian能量问题是图论研究领域的热点问题,最初是由Ivan Gutman、Bo Zhou等人将其概念引入到文献中的。它不仅具有理论价值,更具有重要的实际价值。图的Laplacian能量广泛应用于量子化学、图像层次体系分割、计算机科学等领域。令G= V E)为一个非平凡连通图,边集为E(G),其边数为m,顶点集为V(G),其顶点数为n。若用A(G)表示图G的邻接矩阵,用D(G)表示图G的点的度矩阵,则L(G)=D(G)-A(G)即为图的 Laplacian 矩阵,可以用μi(i=1,2,3...n)表示图的 Laplacian特征值,且μ1≥μ2≥ …μn= 0。则图的Laplacian能量的公式记为LE(G)=∑i=1 n |μi-2m/n|.本文研究了一些应用广泛的特殊图的Laplacian能量的问题,结构及其结果如下:(1)介绍了图论的背景及图的Laplacian能量的基本概念和已有的结论;基于矩阵特征值计算方法,得到了完全图的刺图的Laplacian能量的上下界。(2)利用数学归纳法、分类讨论思想,得到了圈的并图、风车图的Laplacian能量的表达式及其上下界。(3)利用数学归纳法、计算机语言创新序列表示方法、分类讨论、最优化思想,得到了 k-tree 的 Laplacian 能量与 Laplacian 特征值。
[Abstract]:The famous Gneisburg Seven Bridge problem is the origin of graph theory, and then graph theory becomes an important branch of applied mathematics.The Laplacian energy problem of graphs is a hot topic in the field of graph theory, which was first introduced into the literature by Ivan Gutman and Bo Zhou et al.It not only has theoretical value, but also has important practical value.The Laplacian energy of graphs is widely used in quantum chemistry, image hierarchical system segmentation, computer science and so on.Let G = V E) be a nontrivial connected graph, the edge set is an En G ~ (1), the number of edges is m, the vertex set is V _ n G ~ (1), and the number of vertices is n.If the adjoining matrix of graph G is expressed by Agng and the degree matrix of the point of graph G is denoted by DG, then the Laplacian matrix of the graph is the Laplacian matrix of the graph, and the Laplacian eigenvalue of the graph can be expressed by 渭 ~ (1) I ~ (1) (渭 _ (1) 鈮，

本文编号：1715454