Let D be the diameter of a graph G and let λ1 be the largest eigenvalue of its (0,1)adjacency matrix. We give a proof of the fact that there are exactly 69 non-trivial connected graphs with (D + 1)λ1 9. These 69 graphs all have up to 10 vertices and were recently found to be suitable models for small multiprocessor interconnection networks. We also examine the suitability of integral graphs to ...

General statement of the problem. The following algebraic problem is very classical: Let A and B be two Hermitian matrices of the same size. Assume we know the spectrum of each of them. What can be said about the spectrum of their sum A + B? The problem was posed by Weyl [1912]. He gave a series of necessary conditions, known as Weyl’s interlacing inequalities: if λ1(A)≥ · · · ≥ λn(A), λ1(B)≥ ·...

An analytical expression for the maximal Lyapunov exponent λ1 in generalized Fermi-PastaUlam oscillator chains is obtained. The derivation is based on the calculation of modulational instability growth rates for some unstable periodic orbits. The result is compared with numerical simulations and the agreement is good over a wide range of energy densities ε. At very high energy density the power...

Human respiratory syncytial virus (RSV) is a major cause of severe respiratory illness in children and susceptible adults. RSV blocks the development of the innate antiviral immune response and can grow to high titers in the respiratory tract. Here we demonstrate that immunostimulatory defective viral genomes (iDVGs) that are naturally generated during RSV replication are strong inducers of the...

Let F be a nearly holomorphic vector-valued Siegel modular form of weight ρ with respect to some congruence subgroup of Sp2n(Q). In this note, we prove that the function on Sp2n(R) obtained by lifting F has the moderate growth (or “slowly increasing”) property. This is a consequence of the following bound that we prove: ‖ρ(Y )F (Z)‖ ∏n i=1(μi(Y ) λ1/2 + μi(Y ) −λ1/2) where λ1 ≥ . . . ≥ λn is th...

Let AG be the adjacency matrix of G. Let λ1 ≥ λ2 ≥ . . . ≥ λn be the eigenvalues of AG. Sometimes we will also be interested in the Laplacian matrix of G. This is defined to be LG = D−AG, where D is the diagonal matrix where Dvv equals the degree of the vertex v. For d-regular graphs, LG = dI −AG, and hence the eigenvalues of LG are d− λ1, d− λ2, . . . , d− λn. Lemma 1. • λ1 = d. • λ2 = λ3 = . ...

For λ ∈ Z, let λ · A = {λa : a ∈ A}. Suppose r, h ∈ Z are sufficiently large and comparable to each other. We prove that if |A + A| 6 K|A| and λ1, . . . , λh 6 2r, then |λ1 ·A+ . . .+ λh ·A| 6 K |A|. This improves upon a result of Bukh who shows that |λ1 ·A+ . . .+ λh ·A| 6 K|A|. Our main technique is to combine Bukh’s idea of considering the binary expansion of λi with a result on biclique dec...

(1.2) { −∆pu = λa(x)|u|p−2u, u ∈ D 0 (Ω), has the least eigenvalue λ1 > 0 with a positive eigenfunction e1 and λ1 is the only eigenvalue having this property (cf. Proposition 3.1). This gives us a possibility to study the existence of an unbounded branch of positive solutions bifurcating from (λ1, 0). When Ω is bounded, the result is well-known, we refer to the survey article of Amann [2] and t...

For a certain class of sequences with repeated terms, {λn,μn}n=1 := {λ1,λ1, . . . ,λ1 } {{ } μ1 times ,λ2 ,λ2, . . . ,λ2 } {{ } μ2 times , . . . ,λk ,λk , . . . ,λk } {{ } μk times , . . .}, we prove that every function belonging to the closed span of the exponential system {xkeλnx : n ∈ N, k = 0,1,2, . . . ,μn −1}, in some weighted Banach spaces on the real line, extends analytically as an ent...

Among the variety of molecular-graph-based structure-descriptors, aimed at measuring the extent of branching of the skeleton of organic molecules, there are the Wiener index W and the spectral radius λ1 i. e., the greatest eigenvalue of the molecular graph. Although, these indices are used for decades, they have never been compared. Our computational studies show that within series of the isome...