Perturbations on bidegreed graphs minimizing the spectral radius

One common problem in spectral graph theory is to determine which graphs, under some prescribed constraints, maximize or minimize the spectral radius of the adjacency matrix. Here we consider minimizers in the set of bidegreed, or biregular, graphs with pendant vertices and given degree sequence. In this setting, we consider a particular graph perturbation whose effect is to decrease the spectral radius. Hence we restrict the structure of minimizers for k-cyclic degree sequences.     

非正则图的最大特征值

通过对图的最大特征分量与顶点度之间的关系的刻画,得到了图的谱半径与参数最大度和次大度之间的不等关系,进而获得了简单连通非正则图的谱半径的若干上界.     

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     

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.     

谱半径前六位的n阶单圈图

恰含一个圈的简单连通图称为单圈图。Cn记n个顶点的圈。△（i,j,κ）记C3的三个顶点上分别接出i,j,κ条悬挂边所得的图，其中i≥j≥κ≥0.Sl^n-l记Cl的某一顶点上接出n-l条悬挂边所得到的图。△（n-4 1,0,0)记△（n-4,0,0)的某个悬挂点上接出一条悬挂边所得到的图。本文证明了：若把所有n(n≥12）阶单圈图按其最大特征值从大到小的顺序排列，则排在前六位的依次是S3^n-3，△（n-4,1,0),△（n-4 1,0,0),S4^n-4,△（n-5,2,0),△（n-5,1,1)。     

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.     

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.     

On graphs with the largest Laplacian index

Let G be a connected simple graph on n vertices. The Laplacian index of G, namely, the greatest Laplacian eigenvalue of G, is well known to be bounded above by n. In this paper, we give structural characterizations for graphs G with the largest Laplacian index n. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on n and k for the existence of a k-regular graph G of order n with the largest Laplacian index n. We prove that for a graph G of order n ⩾ 3 with the largest Laplacian index n, G is Hamiltonian if G is regular or its maximum vertex degree is Δ(G) = n/2. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results. The first author is supported by NNSF of China (No. 10771080) and SRFDP of China (No. 20070574006). The work was done when Z. Chen was on sabbatical in China.     

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.     

A note on the largest eigenvalue of a large dimensional sample covariance matrix

Let {v_ij; i, J = 1, 2, …} be a family of i.i.d. random variables with E(v₁₁⁴) = ∞. For positive integers p, n with p = p(n) and p/n → y > 0 as n → ∞, let M_n = (1/n) V_n V_n^T , where V_n = (v_ij)_{1 ≤ i ≤ p, 1 ≤ j ≤ n}, and let λ_max(n) denote the largest eigenvalue of M_n. It is shown that a.s. This result verifies the boundedness of E(v₁₁⁴) to be the weakest condition known to assure the almost sure convergence of λ_max(n) for a class of sample covariance matrices.     

Several sharp upper bounds for the largest laplacian eigenvalue of a graph

Let K be the quasi-Laplacian matrix of a graph G and B be the adjacency matrix of the line graph of G,respectively.In this paper,we first present two sharp upper bounds for the largest Laplacian eigenvalue of G by applying the non-negative matrix theory to the similar matrix D~(-1/2) KD~(1/2) and U~(-1/2)BU~(1/2),respectively,where D is the degree diagonal matrix of G and U=diag(d_u,d_v,:uv∈E(G)). And then we give another type of the upper bound in terms of the degree of the vertex and the edge number of G.Moreover,we determine all extremal graphs which achieve these upper bounds.Finally, some examples are given to illustrate that our results are better than the earlier and recent ones in some sense.     

不含三圈的k圈图的拟拉普拉斯和拉普拉斯谱半径

k圈图是边数等于顶点数加k-1的简单连通图.文中确定了不含三圈的k圈图的拟拉普拉斯谱半径的上界,并刻画了达到该上界的极图.此外,文中确定了拟拉普拉斯谱半径排在前五位的不含三圈的单圈图,排在前八位的不含三圈的双圈图.最后说明文中所得结论对不含三圈的k圈图的拉普拉斯谱半径也成立.     

On the spectral spread of bicyclic graphs with given girth

The spectral spread of a graph is defined to be the difference between the largest and the least eigenvalue of the adjacency matrix of the graph. A graph G is said to be bicyclic, if G is connected and |E(G)| = |V(G)|+ 1. Let B(n, g) be the set of bicyclic graphs on n vertices with girth g. In this paper some properties about the least eigenvalues of graphs are given, by which the unique graph with maximal spectral spread in B(n, g) is determined.     

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.     

三圈图的无符号拉普拉斯谱半径

假设图G的点集是V(G)={v_1,v_2,…,v_n},用d_(v_i)(G)表示图G中点v_i的度,令A(G)表示G的邻接矩阵,D(G)是对角线上元素等于d_(v_i)(G)的n×n对角矩阵,Q(G)=D(G)+A(G)是G的无符号拉普拉斯矩阵,Q(G)的最大特征值是G的无符号拉普拉斯谱半径.现确定了所有点数为n的三圈图中无符号拉普拉斯谱半径最大的图的结构.