本文選題：張量 + 譜半徑�。� 參考：《安徽大學(xué)》2015年碩士論文

【摘要】：圖是研究離散對(duì)象二元關(guān)系系統(tǒng)的結(jié)構(gòu)性質(zhì)的一個(gè)重要的數(shù)學(xué)模型.作為圖的推廣,超圖是有限集合的子集系統(tǒng),更加能夠反映現(xiàn)實(shí)世界的對(duì)象之間的復(fù)雜多元關(guān)系.金芳蓉,馮克勤,李文卿,Friedman, Wigderson, Rodriguez,陸臨淵和彭興等引入超圖的鄰接矩陣或Laplace矩陣研究超圖的若干性質(zhì).但是,超圖的每條邊是由兩個(gè)以上的點(diǎn)所確定,所以上述定義沒(méi)有直接地反映超圖的結(jié)構(gòu)性質(zhì).Lek-Heng Lim和祁力群,張恭慶等獨(dú)立地引入了張量的特征值和特征向量及相關(guān)性質(zhì).至此,一致超圖的張量表示(鄰接張量,Laplace張量和無(wú)符號(hào)Laplace張量)的特征值和特征向量得到廣泛研究.2014年,李紅海,祁力群和邵嘉裕討論了樹(shù)的冪圖的最大和最小譜半徑.本文將探討更一般的問(wèn)題,即給定邊數(shù)的一致超圖的最大譜半徑.我們證明了：在給定點(diǎn)數(shù)與邊數(shù)的所有連通的k-一致超圖中,譜半徑達(dá)到最大的圖一定包含一個(gè)與其它所有點(diǎn)都鄰接的點(diǎn).由此,我們分別刻畫(huà)具有最大譜半徑的單圈超圖和給定圍長(zhǎng)的線性或冪單圈(或雙圈)超圖.本文的主要結(jié)構(gòu)如下：在第一章中我們簡(jiǎn)單介紹了圖和超圖譜圖理論的發(fā)展過(guò)程,給出了基本概念和記號(hào),以及本文的研究問(wèn)題和主要結(jié)果。第二章首先介紹非負(fù)張量的Perron-Frobenius定理,然后刻畫(huà)了超圖在其分支遷移后的譜半徑變化情況。第三章刻畫(huà)了具有最大譜半徑的單圈超圖和給定圍長(zhǎng)的線性或冪單圈(或雙圈)超圖.
[Abstract]:Graph is an important mathematical model for studying the structural properties of discrete object binary relation system. As a generalization of graphs, hypergraphs are subsets of finite sets, which can reflect the complex multivariate relationships between objects in the real world. Jin Fangrong, Feng Keqin, Li Wenqing Friedman, Wigderson, Rodriguez, Lu Linyuan and Pengxing introduce adjacent matrices or Laplace matrices to study some properties of hypergraphs. However, each edge of a hypergraph is determined by more than two points, so the above definition does not directly reflect the structural properties of hypergraphs. Lek-Heng Lim and Qili groups. Zhang Gongqing and others independently introduce the eigenvalues, eigenvectors and related properties of Zhang Liang. So far, the eigenvalues and Eigenvectors of the Zhang Liang representations of uniform hypergraphs (adjacent to Zhang Liang Laplace Zhang Liang and unsigned Laplace Zhang Liang) have been extensively studied. In 2014, Li Honghai, Qili Group and Shao Jiayu discussed the maximum and minimum spectral radii of power graphs of trees. In this paper, we will discuss a more general problem, that is, give the maximum spectral radius of a uniform hypergraph with a fixed number of edges. We prove that in all connected k-uniform hypergraphs with a given number of points and edges, a graph with a maximum spectral radius must contain a point adjacent to all other points. Therefore, we characterize the single cycle hypergraphs with the maximum spectral radius and the linear or power unicyclic (or double cycle) hypergraphs with given girth respectively. The main structure of this paper is as follows: in the first chapter, we briefly introduce the development process of graph and hypergraph theory, and give the basic concepts and symbols, as well as the research problems and main results of this paper. In chapter 2, we first introduce the Perron-Frobenius theorem of nonnegative Zhang Liang, and then characterize the variation of spectral radius of hypergraphs after branch migration. In chapter 3, we describe the monocyclic hypergraphs with the maximum spectral radius and the linear or power unicyclic (or double-cycle) hypergraphs with given girth.
【學(xué)位授予單位】：安徽大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類(lèi)號(hào)】：O157.5

【共引文獻(xiàn)】

相關(guān)期刊論文 前5條

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2 Yiyong LI;Qingzhi YANG;Yuning YANG;;A new definition of geometric multiplicity of eigenvalues of tensors and some results based on it[J];Frontiers of Mathematics in China;2015年05期

3 楊志才;裘杭萍;雷智朋;彥杰;;軍事通信網(wǎng)抗毀性研究[J];軍事通信技術(shù);2014年01期

4 鄢仁政;;超圖的子圖特征值的研究[J];海南大學(xué)學(xué)報(bào)(自然科學(xué)版);2014年01期

5 李浙寧;凌晨;王宜舉;楊慶之;;張量分析和多項(xiàng)式優(yōu)化的若干進(jìn)展[J];運(yùn)籌學(xué)學(xué)報(bào);2014年01期

相關(guān)博士學(xué)位論文 前3條

1 王彩玲;連續(xù)與離散最優(yōu)化中的幾個(gè)問(wèn)題[D];吉林大學(xué);2013年

2 楊宇寧;張量特征值與多項(xiàng)式優(yōu)化中的若干問(wèn)題[D];南開(kāi)大學(xué);2013年

3 唐青松;超圖的拉格朗日極值[D];吉林大學(xué);2014年

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本文編號(hào)：2038652