The largest two Laplacian eigenvalues of a graph

In this article, we present lower bounds for the largest eigenvalue, the second largest eigenvalue and the sum of the two largest eigenvalues of the Laplacian matrix of a graph.     

On Trees Whose Second Largest Eigenvalue Does Not Exceed 1

1.IntroductionLetGbeasimplegraphandA(G)betheadjacencymatrixofagraphGwithnvenices.ThecharacteristicpolynomialofGisthecharacteristicpolynomialofitsadjacencymatriX,denotedbyP(G;A),henceP(G;A)=det(AI--A(G)).SinceA(G)isarealsymmeticmatrix,itseigenvaluesmustberealandorderedasThoseAl(A(G))arecalleditheigenvalueofG,denotedAl(A(G))byAl(G).ThereareinseparableconnectionbetweentheconstructionandtheeigenvalueofgraphG.Severalbooksandalotofliterariesoneigenvaluesofgraphsanditsapplicationhavebeenp…     

A note on the second largest eigenvalue of the laplacian matrix of a graph ∗

In this note, a lower bound for the second largest eigenvalue of the Laplacian matrix of a graph is given in terms of the second largest degree of the graph.     

On the extremal values of the second largest Q-eigenvalue

We study extremal graphs for the extremal values of the second largest Q-eigenvalue of a connected graph. We first characterize all simple connected graphs with second largest signless Laplacian eigenvalue at most 3. The second part of the present paper is devoted to the study of the graphs that maximize the second largest Q-eigenvalue. We construct families of such graphs and prove that some of theses families are minimal for the fact that they maximize the second largest signless Laplacian eigenvalue.     

On the second largest eigenvalue of line graphs

In this paper all connected line graphs whose second largest eigenvalue does not exceed 1 are characterized. Besides, all minimal line graphs with second largest eigenvalue greater than 1 are determined. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 61–66, 1998     

A note on the second largest eigenvalue of the laplacian matrix of a graph

In this note, a lower bound for the second largest eigenvalue of the Laplacian matrix of a graph is given in terms of the second largest degree of the graph.     

非负矩阵最大特征值的平滑算法

１引　言 本文中Ａ＝（ａｉｊ）表示ｎ阶方阵,Ａ＞０表示Ａ为正矩阵,即ａｉｊ＞０（ｉ,ｊ＝１,２,…,ｎ）;Ａ≥０表示Ａ为非负矩阵,即ａｉｊ≥０（ｉ,ｊ＝１,２,…,ｎ）且至少有一个严格大于号成立,周知,当Ａ＞０时Ａ有一个正特征值λ满足λ＞｜λ｜,其中λ为Ａ的其它任一特征值;当Ａ≥０时Ａ有一个非负特征值λ满足λ≥｜λ｜,其中λ为Ａ的任一特征值．把这样的λ称为Ａ的最大特征值,为强调它属于Ａ,记作λ（Ａ）．同时,把与λ（Ａ）对应的Ａ的特征向量记作ｘ（Ａ）． 对Ａ≥０,记当Ｒｔ＞０（ｉ＝１,２,…,ｎ）时…     

On graphs whose second largest eigenvalue equals 1 – the star complement technique

The star complement technique is a spectral tool recently developed for constructing some bigger graphs from their smaller parts, called star complements. Here we first identify among trees and complete graphs those graphs which can be star complements for 1 as the second largest eigenvalue. Using the graphs just obtained, we next search for their maximal extensions, either by theoretical means, or by computer aided search.     

On regular graphs and coronas whose second largest eigenvalue does not exceed 1

We characterize all regular graphs whose second largest eigenvalue does not exceed 1. In the sequel, we determine all coronas, different from cones, with the same property. Some results and examples regarding unsolved cases are also given.     

第二大特征值小于$\frac{\sqrt{5}-1}{2}$的平面图

若能将图$G$画在一个平面上,使得任何两条边仅在顶点处相交,则称$G$是平面图.本文刻画了第二大特征值小于$\frac{\sqrt{5}-1}{2}$的所有无孤立点的平面图.     

图的Laplacian特征值

本文研究了连通图的Laplacian特征值,利用图的Laplacian矩阵的特征多项式的行列式表示式,对存在两个不同顶点,但有相同邻集的一类图,得到了一个Laplacian特征值,并给出了它的应用.     

图的无号Laplace矩阵的最大特征值

The signless Laplacian matrix of a graph is the sum of its diagonal matrix of vertex degrees and its adjacency matrix. Li and Feng gave some basic results on the largest eigenvalue and characteristic polynomial of adjacency matrix of a graph in 1979. In this paper, we translate these results into the signless Laplacian matrix of a graph and obtain the similar results.     

树的最大特征值的上界的一个注记

设T是一个树,V是T的顶点集．记dv是υ∈V的度,△是T的最大顶点度．设υ∈V且dw=1．记k=ew+1,这里ew是w的excentricity．设δj′= max{dυ:dist(υ,w)=j},j=1,2,…,k-2,我们证明和这里μ1(T)和λ1(T)分别是T的Laplacian矩阵和邻接矩阵的最大特征值．特别地,记δo′=2．     

On Graphs Whose Spectral Radius is Bounded by \frac{3}{2}\sqrt{2}

The structure of graphs whose largest eigenvalue is bounded by (≈2.1312) is investigated. In particular, such a graph can have at most one circuit, and has a natural quipu structure.     

Multicyclic treelike reflexive graphs

A simple graph is reflexive if its second largest eigenvalue does not exceed 2. A graph is treelike (sometimes also called a cactus) if all its cycles (circuits) are mutually edge-disjoint. In a lot of cases one can establish whether a given graph is reflexive by identifying and removing a single cut-vertex (Theorem 1). In this paper we prove that, if this theorem cannot be applied to a connected treelike reflexive graph G and if all its cycles do not have a common vertex (do not form a bundle), such a graph has at most five cycles (Theorem 2). On the same conditions, in Theorem 3 we find all maximal treelike reflexive graphs with four and five cycles.     

拟树图按第二大无符号拉普拉斯特征值排序

A connected graph G=(V,E) is called a quasi-tree graph if there exists a vertex v_0∈V(G) such that G-v_0 is a tree.In this paper,we determine all quasi-tree graphs of order n with the second largest signless Laplacian eigenvalue greater than or equal to n-3.As an application,we determine all quasi-tree graphs of order n with the sum of the two largest signless Laplacian eigenvalues greater than to 2 n-5/4.     

Spectral characterization of graphs whose second largest eigenvalue is less than 1

Graphs with second largest eigenvalue λ₂?1 are extensively studied, however, whether they are determined by their adjacency spectra or not is less considered. In this paper we completely characterize all the connected bipartite graphs with λ₂<1 that are determined by their adjacency spectra. In addition, we prove that all the connected non-bipartite graphs with girth no less than 4 and λ₂<1 are determined by their adjacency spectra.     

Connected signed graphs of fixed order,size, and number of negative edges with maximal index

In this paper we focus on connected signed graphs of fixed number of vertices, positive edges and negative edges that maximize the largest eigenvalue (also called the index) of their adjacency matrix. In the first step we determine these signed graphs in the set of signed generalized theta graphs. Concerning the general case, we use the eigenvector techniques for getting some structural properties of resulting signed graphs. In particular, we prove that positive edges induce nested split subgraphs, while negative edges induce double nested signed subgraphs. We observe that our concept can be applied when considering balancedness of signed graphs (the property that is extensively studied in both mathematical and non-mathematical context).     

Rellich type identities for eigenvalue problems and application to the Pompeiu problem

For classical Neumann eigenvalue, buckling eigenvalue and clamped plate eigenvalue, we give the corresponding Rellich type identities. As an application of these results, then, we obtain a new necessary and sufficient condition for a domain without the Pompeiu property.     

Non-bipartite Graphs with Third Largest Laplacian Eigenvalue Less Than Three

All bipartite graphs whose third largest Laplacian eigenvalue is less than 3 have been characterized by Zhang. In this paper, all connected non-bipartite graphs with third largest Laplacian eigenvalue less than three are determined.