Markov chains and optimality of the Hamiltonian cycle

We consider the Hamiltonian cycle problem (HCP) embedded in a controlled Markov decision process. In this setting, HCP reduces to an optimization problem on a set of Markov chains corresponding to a given graph. We prove that Hamiltonian cycles are minimizers for the trace of the fundamental matrix on a set of all stochastic transition matrices. In case of doubly stochastic matrices with symmetric linear perturbation, we show that Hamiltonian cycles minimize a diagonal element of a fundamental matrix for all admissible values of the perturbation parameter. In contrast to the previous work on this topic, our arguments are primarily based on probabilistic rather than algebraic methods

NATIONAL BRANDING: THEORETICAL, METHODOLOGICAL AND PRACTICAL ASPECTS (CASE STUDY OF LATVIA, ESTONIA AND REPUBLIC OF MOLDOVA)

In the article are considered theoretical, conceptual and methodological bases of researching national branding. The authors note that, the effective national branding contributes to building an adequate image of the country and acquiring desired positions in the global political system and is aimed at strengthening the global competitiveness of the state and its international political influence.The article also provides a comprehensive analysis of the Latvian, Estonian and Republic of Moldova experience in domain of creation of national brand

Clustering of spectra and fractals of regular graphs

We exhibit a characteristic structure of the class of all regular graphs of degree d that stems from the spectra of their adjacency matrices. The structure has a fractal threadlike appearance. Points with coordinates given by the mean and variance of the exponentials of graph eigenvalues cluster around a line segment that we call a filar. Zooming-in reveals that this cluster splits into smaller segments (filars) labeled by the number of triangles in graphs. Further zooming-in shows that the smaller filars split into subfilars labelled by the number of quadrangles in graphs, etc. We call this fractal structure, discovered in a numerical experiment, a multifilar structure. We also provide a mathematical explanation of this phenomenon based on the Ihara-Selberg trace formula, and compute the coordinates and slopes of all filars in terms of Bessel functions of the first kind.Comment: 10 pages, 5 figure