Which Digraphs with Ring Structure are Essentially Cyclic?

We say that a digraph is essentially cyclic if its Laplacian spectrum is not completely real. The essential cyclicity implies the presence of directed cycles, but not vice versa. The problem of characterizing essential cyclicity in terms of graph topology is difficult and yet unsolved. Its solution is important for some applications of graph theory, including that in decentralized control. In the present paper, this problem is solved with respect to the class of digraphs with ring structure, which models some typical communication networks. It is shown that the digraphs in this class are essentially cyclic, except for certain specified digraphs. The main technical tool we employ is the Chebyshev polynomials of the second kind. A by-product of this study is a theorem on the zeros of polynomials that differ by one from the products of Chebyshev polynomials of the second kind. We also consider the problem of essential cyclicity for weighted digraphs and enumerate the spanning trees in some digraphs with ring structure.Comment: 19 pages, 8 figures, Advances in Applied Mathematics: accepted for publication (2010) http://dx.doi.org/10.1016/j.aam.2010.01.00

Matrices of forests, analysis of networks, and ranking problems

The matrices of spanning rooted forests are studied as a tool for analysing the structure of networks and measuring their properties. The problems of revealing the basic bicomponents, measuring vertex proximity, and ranking from preference relations / sports competitions are considered. It is shown that the vertex accessibility measure based on spanning forests has a number of desirable properties. An interpretation for the stochastic matrix of out-forests in terms of information dissemination is given.Comment: 8 pages. This article draws heavily from arXiv:math/0508171. Published in Proceedings of the First International Conference on Information Technology and Quantitative Management (ITQM 2013). This version contains some corrections and addition

Ethnonyms in Karabakh Folklore and their Linguistic Characteristics

Onomastic units in the language are related to folklore and all spiritual values. Language is not part of a culture, but it accumulates cultural ideas throughout its history, the world and human concepts through linguistic means, in other words, in words, phrases, specific lexical layers, and sometimes even in grammar. Language serves as the material of cultural expression while being the meta-language of culture, which fixes the meanings, motives and interpretations of oral texts in verbal forms. Viewing language as a source of cultural codes, forms of expression of cultural tradition, study and reference for regeneration of ancient culture, along with folklore texts, ceremonies, and folk, would be more accurate.The ancient history and geography of the people live in the language. The names of ancient tribes, clans and peoples that existed in the homeland of the language speakers have been preserved in folklore examples and have survived to this day. Tribe, folk names - ethnonyms are onomastic units frequently in Karabakh folklore.Information about Azerbaijan's ancient and medieval ethnonyms can be found mainly in the works of ancient authors and Persian-language texts from the medieval period.The study of ethnonyms is also helpful from the historical point of view of the Azerbaijani language. The phonetic differences between the written forms of ethnonyms in the sources and their pronounced documents in our toponymy are significant.The article is based on examples collected and compiled from the book of 10 volumes, "Karabakh: folklore is also a History", published by the Institute of Folklore of the Azerbaijan National Academy of Sciences. The etymology of ethnonyms used in Karabakh folklore has been reviewed