The k-th spectral moment Mk(G) of the adjacency matrix a graph G represents number closed walks length k in G. We study here partial order ≼ graphs, defined by H if ≤ Mk(H) for all ≥ 0, and are interested question when is linear within specified set graphs? Our main result that on each starlike trees with constant vertices. Recall connected tree it has vertex u such components − paths, called b...

We have developed simple algorithms that allow adjacency and proximity searching in Google and the Science Citation Index (SCI). The SCI algorithm exploits the fact that SCI stopwords in a search phrase function as a placeholder. Such a phrase serves effectively as a fixed adjacency condition determined by the number n of adjacent stopwords (i.e., retrieve all records where word A and word B ar...

We describe how to create a continuous global index of the surface of the earth. The model is based on a hierarchical subdivision of the surface into triangular regions, in which each region is assigned a numerical label according to a space lling curve. Sequential labels are assigned to adjacent regions, so labels can be sorted to create a continuous one-dimensional index. Bene ts of this cont...

The purpose of this paper is to discuss spectral bounds on the chromatic number of a graph. The classic result by Hoffman, where λ1 and λn are respectively the maximum and minimum eigenvalues of the adjacency matrix of a graph G, is χ(G) ≥ 1− λ1 λn . It is possible to discuss the coloring of Hermitian matrices in general. Nikiforov developed a spectral bound on the chromatic number of such matr...

In this work we present a dynamical model that succesfully describes the organization of mutualistic ecological systems. The main characteristic of these systems is the nested structure of the bipartite adjacency matrix describing their interactions. We introduce a nestedness coefficient, as an alternative to the Atmar and Patterson temperature, commonly used to measure the nestedness degree of...

The index of a signed graph is the largest eigenvalue its adjacency matrix. For positive integers n and m ? 2 / 4 , we determine maximum complete graphs with vertices negative edges characterize achieving this maximum. This settles (the corrected version of) conjecture by Koledin Stani? (2017).

We study space and time efficient quantum algorithms for two graph problems – deciding whether an n-vertex graph is a forest, and whether it is bipartite. Via a reduction to the s-t connectivity problem, we describe quantum algorithms for deciding both properties in Õ(n) time and using O(log n) classical and quantum bits of storage in the adjacency matrix model. We then present quantum algorith...

We study the effects of network topology on the response of networks of coupled discrete excitable systems to an external stochastic stimulus. We extend recent results that characterize the response in terms of spectral properties of the adjacency matrix by allowing distributions in the transmission delays and in the number of refractory states and by developing a nonperturbative approximation ...

Estimating and minimizing the maximum power dissipation during testing is an important task in VLSI circuit realization since the power value affects the reliability of the circuits. Therefore during testing a methodology should be adopted to minimize power consumption. Test patterns generated with –D 1 option of ATALANTA contains don’t care bits (x bits). By suitable filling of don’t cares can...