In this paper, at first we introduce a new index with the name Co-PI index and obtain some properties related this new index. Then we compute this new index for TUC4C8(R) nanotubes.

Given a set of points P = {p1, p2 . . . , pn} ⊂ R, the 1-center problem is to determine the smallest radius ball containing P . Let x be the center of the ball and let r be its radius. We can express the problem of finding the 1-center as the following optimization problem. min r, ||x− pi|| ≤ r ∀i ≤ n In order to avoid computing square roots, we can reformulate the problem as: min r ||x− pi|| ≤...

Aims. We investigate the physical properties and spatial distribution of Carbon Monoxide (CO) gas in the disks around the Herbig Ae/Be stars HD 97048 and HD 100546. Methods. Using high-spectral-resolution 4.588−4.715 μm spectra containing fundamental CO emission taken with CRIRES on the VLT, we probe the circumstellar gas and model the kinematics of the emission lines. By using spectro-astromet...

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 ...

Let G be a simple undirected graph. For v ∈ V (G), the 2-degree of v is the sum of the degrees of the vertices adjacent to v. Denote by ρ(G) and μ(G) the spectral radius of the adjacency matrix and the Laplacian matrix of G, respectively. In this paper, we present two lower bounds of ρ(G) and μ(G) in terms of the degrees and the 2-degrees of vertices. © 2004 Elsevier Inc. All rights reserved. A...

Given a graph G, write μ (G) for the largest eigenvalue of its adjacency matrix, ω (G) for its clique number, and wk (G) for the number of its k-walks. We prove that the inequalities wq+r (G) wq (G) ≤ μ (G) ≤ ω (G) − 1 ω (G) wr (G) hold for all r > 0 and odd q > 0. We also generalize a number of other bounds on μ (G) and characterize pseudo-regular and pseudo-semiregular graphs in spectral terms.

Let G be a graph, χ be its chromatic number, λ be the largest eigenvalue of its Laplacian, and µ be the largest eigenvalue of its adjacency matrix. Then, complementing a well-known result of Hoffman, we show that λ ≥ χ χ − 1 µ with equality holding for regular complete χ-partite graphs. We denote the eigenvalues of a Hermitian matrix A as µ (A) = µ 1 (A) ≥ · · · ≥ µ min (A). Given a graph G, we...

We characterize the graphs which achieve the maximum value of the spectral radius of the adjacency matrix in the sets of all graphs with a given domination number and graphs with no isolated vertices and a given domination number. AMS Classification: 05C35, 05C50, 05C69