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Представляем Научные Достижения Миру. Естественные Науки Presenting Academic Achievements to the World Представляем научные достижения миру. Естественные науки Presenting Academic Achievements to the World. Natural Science Papers from the IX conference for young scientists «Presenting Academic Achievements to the World» April 10–11, 2018 Saratov Issue 8 Saratov 2019 Представляем научные достижения миру. Естественные науки Материалы IX научной конференции молодых ученых «Presenting Academic Achievements to the World» Апрель 10–11, 2018 Саратов Выпуск 8 Саратов 2019 УДК 5(082) ББК 20я43 П71 Представляем научные достижения миру. Естественные П71 науки : материалы IX научной конференции молодых ученых «Presenting Academic Achievements to the World». – Саратов: Изд-во «Саратовский источник», 2019. – Вып. 8. – 224 с. : ил. В данном сборнике опубликованы материалы участников секции естественных наук IX научной конференции молодых ученых «Presenting Academic Achievements to the World», которая состоялась в Саратовском государственном университете 10–11 апреля 2018 года. В сборник включены статьи с результатами исследований в области физики, химии, географии, геологии и информационных технологий. This publication assembles papers given at the IX conference for young scientists «Presenting Academic Achievements to the World» which was held in April 10–11, 2018 at Saratov State University. The articles present the results in such fi elds of natural science as Physics, Chemistry, Geography, Geology and Information Technology. Редакционная коллегия: С. А. Шилова (отв. редактор), Л. В. Левина (отв. секретарь), Д. А. Алексеева, А. А. Сосновская, С. В. Пыжонков УДК 5(082) ББК 20я43 Работа издана в авторской редакции ISSN 2306-3068 ABOUT GENERATION OF NON-ISOMORPHIC VERTEX K-COLORINGS BY MCKAY’S APPROACH M. B. Abrosimov, P. V. Razumovsky Saratov State University Abstract: We consider generation of all non-isomorphic vertex k-colorings prob- lem. This article contains resulting algorithm for generation colorings without isomor- phism proving by McKay’s approach. Problem of all non-isomorphic edge k-colorings generation reduced to vertex colorings generation problem. Keywords: coloring, graph coloring, graph, isomorphism. О ГЕНЕРАЦИИ НЕИЗОМОРФНЫХ ВЕРШИННЫХ K-РАСКРАСОК МЕТОДОМ МАККЕЯ М. Б. Абросимов, П. В. Разумовский Саратовский национальный исследовательский государственный университет имени Н. Г. Чернышевского Аннотация: В данной работе рассматривается задача генерации всех неизо- морфных вершинных k-раскрасок. Предоставляется алгоритм генерации раскра- сок без проверки на изоморфизм методом МакКея. Задача генерации всех неизо- морфных реберных раскрасок сводится к задаче генерации вершинных раскрасок. Ключевые слова: раскраска, граф, изоморфизм. INTRODUCTION Defi nition 1. Let – graph, . Function called vertex k-coloring of the graph, – vertex color. Graph G, where each vertex has it’s own color, called colored graph, or graph with colored vertices. Let denote such graphs as . 5 Edge k-coloring defi nes similarly. Defi nition 2. Graph color isomorphism for two graphs and is an isomorphism of and which keeps graphs vertices’ colors in place. This is a bijection , which satisfi es the followingg conditions: 1. ; 2. Defi nition 3. Graph color automorphism – an isomorphism of colored graph on itself. A set of all color automorphisms, including asymmetric auto- morphism, forms graph’s automorphism group. Defi nition 4. Two vertices called similar, if there is the automorphism, which maps the fi rst vertex to the second. A set of all similar vertices called an orbit. NON-ISOMORPHIC COLORED GRAPHS CONSTRUCTION BY MCKAY’S APPROACH We can reduce the problem of generating all non-isomorphic edge color- ings of the defi ned graph to the problem of generating vertex colorings by edge graph construction. If two graphs and are isomorphic then two edge graphs and are isomorphic too. It is known that the suffi cient case is valid for all graphs except only one pair of non-isomorphic graphs and , which have the same edge graph. So, if we construct edge graph and consider the special case, we can solve edge colorings problem. So, generating all non-isomorphic vertex colorings is the main problem, which has no effective solution. There are several approaches, the most effec- tive is the isomorphism rejection technique. It allows to get rid of isomorphism proving (description of this technique could be found in (Brinkmann, 2000)). The base of the technique is to construct canonical labeling: 1. Defi ne rules of calculating graph labeling. 2. Select canonical labeling from the set of all graph labelings. 3. Generate all graphs with their labelings. 4. Accept graphs, which have canonical labelings, reject other. Such approach has been investigated in the article (Абросимов, Разумовский, 2017). In this article we consider McKay’s approach. McKay’s approach establishes the order of generating structures itera- tively from small to large: from an ancestor to a descendant. One structure could have several ancestors at one time. 6 McKay implements isomorphism rejection the next way (McKay, 1998): 1. Defi ne one canonical ancestor for each structure and a way, how this structure could be constructed. 2. Accept structure if and only if it generates by the right way from the canonical ancestor. 3. Each ancestor generates all non-isomorphic descendant only once. COLORING GENERATION RULES Descendant generation is produced by calculating orbits for colored graph. Graph coloring specifi es vertices partition. Partition allows to build au- tomorphisms group. Then we can calculate the set of graph orbits. After that select one member from each orbit and paint in all possible colors. The resulted set of colorings is a set of descendants. When we build all colorings, we need to validate that each descendant has a canonical ancestor: iteratively remove one vertex color and compare the current coloring with the ancestor. If the ancestor is greater than the coloring then the ancestor is not canonical, we need to reject the descendant. Otherwise the descendant has a canonical ancestor. ALGORITHM OF GENERATING ALL NON-ISOMORPHIC VERTEX K-COLORINGS Automorphisms group building and orbits calculating have been imple- mented with using nauty program (McKay, 2017; McKay, Piperno,p) 2013). Let specify the graphp and number of colors . 1. Defi ne vector – initializing vector of graph’s vertices colors, where is a color of i-th vertex. 2. Build automorphisms group and calculate a set of orbits for the current coloring . 3. Select one member o from each orbit ((let select maximal member) and iteratively paint in each color from , build coloring’s descendants child. 4. Validate each child, that has the canonical descendant by the defi ned way, described above. 5. For each accepted child generation returned to the second step. All generated colorings satisfi es the condition of lexicographic minimum and non-isomorphic between each other. Isomorphism proving is not used in the algorithm. 7 REFERENCES 1. Абросимов М. Б., Разумовский П. В. О генерации неизоморфных вершинных k-раскрасок // Прикладная дискретная математика. Приложение. 2017. № 10. С. 136–138. 2. Brinkmann G. Isomorphism rejection in structure generation programs // Discrete Math- ematical Chemistry, DIMACS Series in Discrete Mathematics and Theoretical Com- puter Science. 2000. Vol. 51. P. 25–38. 3. McKay B. D. Isomorph-free exhaustive generation // Journal of Algorithms. 1998. Vol. 26. Pp. 306–324. 4. McKay B. D. nauty and Traces: Graph canonical labeling and automorphism group com- putation // 2017. URL: http://users.cecs.anu.edu.au/~bdm/nauty/nug26.pdf. 5. McKay B. D., Piperno A. Practical Graph Isomorphism // J. Symbolic Computation. 2013. Vol. 2, 60. Pp. 94–112. COMBINED REACTION-RECTIFICATION PENTANE-HEXANE FRACTION ISOMERIZATION PROCESS A. A. Amikishiev Saratov State University Abstract: The article analyses several types of combined isomerization-rectifi ca- tion process. The purpose of this work is to choose the optimal technological scheme for the isomerization of the pentane-hexane fraction in reaction-rectifi cation column. Keywords: isomerization, combined process, pentane-hexane fraction, octane number. СОВМЕЩЕННЫЙ РЕКЦИОННО-РЕКТИФИКАЦИОННЫЙ ПРОЦЕСС ИЗОМЕРИЗАЦИИ ПЕНТАН-ГЕКСАНОВОЙ ФРАКЦИИ А. А. Амикишиев Саратовский национальный исследовательский государственный университет имени Н. Г. Чернышевского Аннотация: Статья включает анализ нескольких видов совмещенных ре- акционно-ректификационных процессов. Целью данной работы является выбор оптимальной технологической схемы для процесса изомеризации пентан-гекса- новой фракции в реакционно-ректификационной колонне. Ключевые слова: Изомеризация, совмещенный процесс, пентан-гексановая фракция, октановое число. 8 The major need of the izomerization processes development is deter- mined by the adoption of the more strict operating procedures that stiffen the technological requirements to the motor gasolines, including the content of aromatics, particularly benzole, limits (Кузьмина, Фролов, Ливенцев, 2008). Nevertheless, the contemporary technologies, though providing high quality of isomerizate are marked by the critical fl aws, such as need of bulky and complex recirculation systems, which usage is induced by the low con- version rates of a single pass scheme. In the meantime, it is known that the combined processes allow accomplishing high crude conversion rates directly in the reaction-mass-transfer apparatus, thus cutting down the number and ca- pacity of external recycles and reducing the catalyst volume. The objective of the research has been to choose a technological scheme for the
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