图能量与边控制数关系的研究
发布时间：2018-08-03 14:17
【摘要】：对于一个无环无重边的简单图G,分别用V(G)和E(G)来表示图G的点集和边集.如果E(G)的子集F满足不在集合F中的任意一条边都至少与F中的一条边相邻接,则F称为G的边控制集.G的最小边控制集中所包含的边的条数称为图G的边控制数,记作θ(G).图G的能量ε(G)是G的所有特征值的绝对值之和.我们知道图的一个代数不变量——图的能量在图理论中占据重要地位,它在物理、化学等领域也有着广泛应用.Gutman将能量的概念推广到所有简单图,他定义简单图G的能量为ε(G)=(?),其中λ1,...,λn是G特征值.显然,如果我们能计算出一个图的特征值,我们就能立刻知道它的能量.但计算大规模矩阵的特征值是非常困难的,即使对于像邻接矩阵A(G)这样的(0,1)-对称矩阵也是十分困难的.于是,许多研究者便对某些图类建立了很多能量的上、下界来估计这一不变量.本文主要研究图G的能量与边控制数之间的关系.主要内容如下:第一章介绍与图的能量有关的研究背景和现状.第二章介绍了与本文有关的概念和已知的结论.第三章我们研究一种特殊情况——图的控制集为一条边时图的能量.第四章用先用图的边控制数证明了图能量的下界.如果G边控制数为θ的连通图,则ε(G)≥ 2θ,等号成立当且仅当G是完全二部图Kθ.θ.接着用图的边控制数证明了图能量的上界.ε(G)≤2θ(?)+(θ2-θ)((?)+ 1)Δ上界可达当且仅当G是由一条边连接两个K1,Δ-1的中心点得到的图形,其中Δ是G的顶点的最大度.
[Abstract]:For a simple graph G with no ring and no repeated edges, the point set and edge set of graph G are represented by V (G) and E (G), respectively. If the subset F of E (G) satisfies any edge adjacent to at least one of the edges in the set F, then F is called the edge control set of G. the number of edges contained in the minimum edge control set of G. is called the edge domination number of graph G. The energy 蔚 (G) of graph G is the sum of the absolute values of all eigenvalues of G. We know that an algebraic invariant of a graph the energy of a graph plays an important role in graph theory, and that it is also widely used in physics, chemistry, and so on. Gutman extends the concept of energy to all simple graphs. He defines the energy of a simple graph G as 蔚 (G) = (?), where 位 1n is the eigenvalue of G. Obviously, if we can calculate the eigenvalue of a graph, we can immediately know its energy. However, it is very difficult to calculate the eigenvalues of large-scale matrices, even for (0 ~ 1) -symmetric matrices such as the adjacent matrix A (G). Therefore, many researchers have established a lot of upper and lower bounds of energy for some graph classes to estimate this invariant. In this paper, we study the relationship between the energy of graph G and the edge domination number. The main contents are as follows: the first chapter introduces the research background and current situation of energy related to graphs. The second chapter introduces the concepts and known conclusions related to this paper. In chapter 3, we study the energy of a special case in which the control set of a graph is an edge timed graph. In chapter 4, the lower bound of graph energy is proved by using the edge domination number of graph. If the G edge domination number is a connected graph with 胃, then 蔚 (G) 鈮，
本文编号：2162007
[Abstract]:For a simple graph G with no ring and no repeated edges, the point set and edge set of graph G are represented by V (G) and E (G), respectively. If the subset F of E (G) satisfies any edge adjacent to at least one of the edges in the set F, then F is called the edge control set of G. the number of edges contained in the minimum edge control set of G. is called the edge domination number of graph G. The energy 蔚 (G) of graph G is the sum of the absolute values of all eigenvalues of G. We know that an algebraic invariant of a graph the energy of a graph plays an important role in graph theory, and that it is also widely used in physics, chemistry, and so on. Gutman extends the concept of energy to all simple graphs. He defines the energy of a simple graph G as 蔚 (G) = (?), where 位 1n is the eigenvalue of G. Obviously, if we can calculate the eigenvalue of a graph, we can immediately know its energy. However, it is very difficult to calculate the eigenvalues of large-scale matrices, even for (0 ~ 1) -symmetric matrices such as the adjacent matrix A (G). Therefore, many researchers have established a lot of upper and lower bounds of energy for some graph classes to estimate this invariant. In this paper, we study the relationship between the energy of graph G and the edge domination number. The main contents are as follows: the first chapter introduces the research background and current situation of energy related to graphs. The second chapter introduces the concepts and known conclusions related to this paper. In chapter 3, we study the energy of a special case in which the control set of a graph is an edge timed graph. In chapter 4, the lower bound of graph energy is proved by using the edge domination number of graph. If the G edge domination number is a connected graph with 胃, then 蔚 (G) 鈮，
本文编号：2162007
本文链接：https://www.wllwen.com/kejilunwen/yysx/2162007.html
