On graphs with given main eigenvalues

An eigenvalue of a graph is main if it has an eigenvector, the sum of whose entries is not equal to zero. Extending previous results of Hagos and Hou et al. we obtain two conditions for graphs with given main eigenvalues. All trees and connected unicyclic graphs with exactly two main eigenvalues were characterized by Hou et al. Extending their results, we determine all bicyclic connected graphs with exactly two main eigenvalues.     

Tricyclic graphs with exactly two main eigenvalues

An eigenvalue of a graph G is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Let G ₀ be the graph obtained from G by deleting all pendant vertices and δ(G) the minimum degree of vertices of G. In this paper, all connected tricyclic graphs G with δ(G ₀) ≥ 2 and exactly two main eigenvalues are determined.     

恰有两个Q-主特征值的三圈图

图G的无符号拉普拉斯矩阵定义为图G的邻接矩阵与度对角矩阵的和,其特征值称为图G的Q-特征值.图G的一个Q-特征值称为Q-主特征值,如果它有一个特征向量其分量的和不等于零.确定了所有恰有两个Q-主特征值的三圈图.     

The integral graphs with index 3 and exactly two main eigenvalues

A graph is called integral if the spectrum of its adjacency matrix has only integral eigenvalues. An eigenvalue of a graph is called main eigenvalue if it has an eigenvector such that the sum of whose entries is not equal to zero. In this paper, we show that there are exactly 25 connected integral graphs with exactly two main eigenvalues and index 3.     

Main eigenvalues of a graph

An eigenvalue of a graph is said to be a main eigenvalue if it has an eigenvector not orthogonal to the main vector j=(1,1,…,1). In this paper we shall study some properties of main eigenvalues of a graph.     

Trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues

The signless Laplacian matrix of a graph G is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called Q-eigenvalues of G. A Q-eigenvalue of a graph G is called a Q-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this work, all trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues are determined.     

恰含两个圈的二部图的能量

设λ1,λ2,…,λn是n阶图G的特征值,图G的能量是E(G)=|λ1| |λ2| … |λn|,设G(n)是n个顶点n 1条边的恰有两个圈的连通二部图的集合,Z(n;4,4)是G(n)中的一个图,它的两个长为4的圈恰有一个公共点,其余n-7个点都是悬挂点且均与这个公共点相邻．文中证明了Z(n;4,4)是G(n)中具有最小能量的图。     

双圈图的最大Estrada指数

令Bn^＋表示顶点个数为礼的双圈二部图的集合．考虑了召吉中图依Estrada指数从大到小的排序问题．利用二部图的Estrada指数和最大特征值之间的关系,当n≥8时,得到了Bn^＋中具有最大和次大Estrada指数的图．     

On bipartite graphs with small number of laplacian eigenvalues greater than two and three

All connected bipartite graphs with exactly two Laplacian eigenvalues greater than two are determined. Besides, all connected bipartite graphs with exactly one Laplacian eigenvalue greater than three are determined.     

On bipartite graphs with small number of laplacian eigenvalues greater than two and three

All connected bipartite graphs with exactly two Laplacian eigenvalues greater than two are determined. Besides, all connected bipartite graphs with exactly one Laplacian eigenvalue greater than three are determined.     

A Characterization of Distance-Regular Graphs with Diameter Three

We characterize the distance-regular graphs with diameter three by giving an expression for the number of vertices at distance two from each given vertex, in terms of the spectrum of the graph.     

On signed graphs with just two distinct adjacency eigenvalues

In this paper, all simple connected signed graphs with maximum degree at most 4 and with just two distinct adjacency eigenvalues are completely characterized, there exists an infinite family of 4-regular signed graphs with just two distinct adjacency eigenvalues.