Coarse non-amenability and covers with small eigenvalues

Given a closed Riemannian manifold M and a (virtual) epimorphism ${\pi_1(M)\twoheadrightarrow \mathbb{F}_2}$ of the fundamental group onto a free group of rank 2, we construct a tower of finite sheeted regular covers ${\left\{M_n\right\}_{n=0}^{\infty}}$ of M such that ?? ₁(M _n ) ?? 0 as n ?? ??. This is the first example of such a tower which is not obtainable up to uniform quasi-isometry (or even up to uniform coarse equivalence) by the previously known methods where ?? ₁(M) is supposed to surject onto an amenable group.     

Universal adjacency matrices with two eigenvalues

Consider a graph Γ on n vertices with adjacency matrix A and degree sequence (d₁,…,d_n). A universal adjacency matrix of Γ is any matrix in Span {A,D,I,J} with a nonzero coefficient for A, where and I and J are the n×n identity and all-ones matrix, respectively. Thus a universal adjacency matrix is a common generalization of the adjacency, the Laplacian, the signless Laplacian and the Seidel matrix. We investigate graphs for which some universal adjacency matrix has just two eigenvalues. The regular ones are strongly regular, complete or empty, but several other interesting classes occur.     

Bicyclic 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, and it is well known that a graph has exactly one main eigenvalue if and only if it is regular. In this work, all connected bicyclic graphs with 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.     

Nonnegatively realizable spectra with two positive eigenvalues

Boyle and Handelman [M. Boyle and D. Handelman, The spectra of nonnegative matrices via symbolic dynamics, Ann. Math. 133 (1991), pp. 249–316.] characterized all lists of n complex numbers that can be the nonzero spectrum of a nonnegative matrix. This article presents a constructive proof of this result in the special case when the lists are real and contain two positive numbers and n ? 2 negative numbers. A bound for the number of zeros that needs to be added to the list to achieve a nonnegative realization is presented in this case.     

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.     

Antipodal distance-transitive covers with primitive quotient of diameter two

As a consequence of a famous theorem by Derek Smith, an unknown distance-transitive graph is either primitive of diameter at least two and valency at least three or is antipodal, bipartite, or both. In the imprimitive cases an unknown graph must have a primitive core of diameter at least two and valency at least three. It seems that the known list of primitive graphs is complete. Here, starting from an earlier work by Brouwer and Van Bon, we find every distance-transitive antipodal cover whose primitive quotient is one of the known distance-transitive graphs of diameter two and valency at least three.     

Products of two involutions with prescribed eigenvalues and some applications

Let F be a field. In [Djokovic, Product of two involutions, Arch. Math. 18 (1967) 582-584] it was proved that a matrix A∈F^n×n can be written as A=BC, for some involutions B,C∈F^n×n, if and only if A is similar to A^-1. In this paper we describe the possible eigenvalues of the matrices B and C.As a consequence, in case charF≠2, we describe the possible similarity classes of (P₁₁⊕P₂₂)P^-1, when the nonsingular matrix P=[P_ij]∈F^n×n, i,j∈{1,2} and P₁₁∈F^s×s, varies.When F is an algebraically closed field and charF≠2, we also describe the possible similarity classes of [A_ij]∈F^n×n, i,j∈{1,2}, when A₁₁ and A₂₂ are square zero matrices and A₁₂ and A₂₁ vary.     

Computational aspect to the nearest matrix with two prescribed eigenvalues

Given a complex square matrix A and two complex numbers λ₁ and λ₂, we present a method to calculate the distance from A to the set of matrices X that have λ₁ and λ₂ as some of their eigenvalues. We also find the nearest matrix X.     

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.     

Invariants of linear groups generated by matrices with two nonunit eigenvalues

A theorem on the structure of the algebra of invariants of the commutant of a group generated by pseudoreflections is improved. In particular, it is shown that this algebra is a complete intersection. A series of counterexamples to Stanley's conjecture is constructed in dimension 4. Results supporting this conjecture for primitive groups of large dimension are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 120–130, 1982.     

The determinant of the sum of two normal matrices with prescribed eigenvalues

Given complex numbers α₁,...,α_n, β₁,...,β_n, what can we say about the determinant of A+B, where A (B) is an n×n normal matrix with eigenvalues α₁,...,α_n (β₁,...,β_n)? Some partial answers are offered to this question.     

The graphs with all but two eigenvalues equal to $$$$ or 0

We determine all graphs for which the adjacency matrix has at most two eigenvalues (multiplicities included) not equal to \(-2\), or 0, and determine which of these graphs are determined by their adjacency spectrum.     

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.     

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.     

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.     

Local finitely smooth equivalence of real autonomous systems with two pure imaginary eigenvalues

The paper deals with real autonomous systems of ordinary differential equations in a neighborhood of a nondegenerate singular point such that the matrix of the linearized system has two pure imaginary eigenvalues, all other eigenvalues lying outside the imaginary axis. It is proved that, for such systems having a focus on the center manifold, the problem of finitely smooth equivalence is solved in terms of the finite segments of the Taylor series of their right-hand sides.