【摘要】：對于一個無環(huán)無重邊的簡單圖G,分別用V(G)和E(G)來表示圖G的點集和邊集.如果E(G)的子集F滿足不在集合F中的任意一條邊都至少與F中的一條邊相鄰接,則F稱為G的邊控制集.G的最小邊控制集中所包含的邊的條數(shù)稱為圖G的邊控制數(shù),記作θ(G).圖G的能量ε(G)是G的所有特征值的絕對值之和.我們知道圖的一個代數(shù)不變量——圖的能量在圖理論中占據(jù)重要地位,它在物理、化學(xué)等領(lǐng)域也有著廣泛應(yīng)用.Gutman將能量的概念推廣到所有簡單圖,他定義簡單圖G的能量為ε(G)=(?),其中λ1,...,λn是G特征值.顯然,如果我們能計算出一個圖的特征值,我們就能立刻知道它的能量.但計算大規(guī)模矩陣的特征值是非常困難的,即使對于像鄰接矩陣A(G)這樣的(0,1)-對稱矩陣也是十分困難的.于是,許多研究者便對某些圖類建立了很多能量的上、下界來估計這一不變量.本文主要研究圖G的能量與邊控制數(shù)之間的關(guān)系.主要內(nèi)容如下:第一章介紹與圖的能量有關(guān)的研究背景和現(xiàn)狀.第二章介紹了與本文有關(guān)的概念和已知的結(jié)論.第三章我們研究一種特殊情況——圖的控制集為一條邊時圖的能量.第四章用先用圖的邊控制數(shù)證明了圖能量的下界.如果G邊控制數(shù)為θ的連通圖,則ε(G)≥ 2θ,等號成立當(dāng)且僅當(dāng)G是完全二部圖Kθ.θ.接著用圖的邊控制數(shù)證明了圖能量的上界.ε(G)≤2θ(?)+(θ2-θ)((?)+ 1)Δ上界可達當(dāng)且僅當(dāng)G是由一條邊連接兩個K1,Δ-1的中心點得到的圖形,其中Δ是G的頂點的最大度.
[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，

本文編號：2162007