In this paper, we show the evaluation of the spectral radius for node degree as the basis to analyze the variation in the node degrees during the evolution of scale-free networks and small-world networks. Spectral radius is the principal eigenvalue of the adjacency matrix of a network graph and spectral radius ratio for node degree is the ratio of the spectral radius and the average node degree...

The p-spectral radius of a uniform hypergraph G of order n is de…ned for every real number p 1 as (p) (G) = max jx1j + + jxnj = 1 r! X fi1;:::;irg2E(G) xi1 xir : It generalizes several hypergraph parameters, including the Lagrangian, the spectral radius, and the number of edges. The paper presents solutions to several extremal problems about the p-spectral radius of k-partite and k-chromatic hy...

This paper deals with the joint spectral radius of a finite set of matrices. We say that a set of matrices has the finiteness property if the maximal rate of growth, in the multiplicative semigroup it generates, is given by the powers of a finite product. Here we address the problem of establishing the finiteness property of pairs of 2× 2 sign-matrices. Such problem is related to the conjecture...

We provide upper estimates on the spectral radius of a directed graph. In particular we prove that the spectral radius is bounded by the maximum of the geometric mean of in-degree and out-degree taken over all vertices.

In this paper, we study spectral versions of the densest subgraph problem and the largest independence subset problem. In the first part, we give an algorithm for identifying small subgraphs with large spectral radius. We also prove a Hoffman-type ratio bound for the order of an induced subgraph whose spectral radius is bounded from above.

In this paper, we give the spectral theory for eigenvalues and eigenfunctions of a boundary value problem consisting of the linear fractional Bessel operator. Moreover, we show that this operator is self-adjoint, the eigenvalues of the problem are real, and the corresponding eigenfunctions are orthogonal. In this paper, we give the spectral theory for eigenvalues and eigenfunctions...

Echo State Networks (ESN) are reservoir networks that satisfy well-established criteria for stability when constructed as feedforward networks. Recent evidence suggests that stability criteria are altered in the presence of reservoir substructures, such as clusters. Understanding how the reservoir architecture affects stability is thus important for the appropriate design of any ESN. To quantit...

Weighted automata over the tropical semiring Zmax = (Z ∪ {−∞},max,+) are closely related to finitely generated semigroups of matrices over Zmax. In this paper, we use results in automata theory to study two quantities associated with sets of matrices: the joint spectral radius and the ultimate rank. We prove that these two quantities are not computable over the tropical semiring, i.e. there is ...