Матричне подання операцiй над графами
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Дата
2022
Науковий керівник
Укладач
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E-ISSN
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Анотація
Теорiя графiв має широке розповсюдження з практичної точки зору. Графи оточують нас у повсякденному життi (наприклад, карти дорiг та шляхiв), а також вiдiграють важливу роль в наукових дослiдженнях (наприклад, електросхеми). Для побутового застосування, безумовно, найзручнiшим є геометричний спосiб подання графiв. Але для комп’ютерної обробки iнформацiї це не є рацiональним. В цих випадках використовується матричне подання графiв у виглядi матриць сумiжностi або матриць iнцидентностi. Тому все бiльшого значення набувають дослiдження, присвяченi саме цiй темi. В данiй статтi розглядається можливiсть виконання операцiй над матрицями, якими подано графи. Цi методи мають свої особливостi та обмеження. Вони також розглянутi в данiй статтi. Для кожної операцiй запропонований варiант обробки як матрицi сумiжностi, так i матрицi iнцидентностi для орiєнтованих та неорiєнтованих графiв, показано вiдмiнностi такої обробки в залежностi вiд виду графу. MSC: 03G05, 03G25, 03F52, 06E25, 15B34.
This article considers the possibility of matrix execution of both unary and binary operations on graphs. Graphs, as an abstract mathematical construction, have a very wide range of practical applications. First of all, it is algorithmization and computer processing of information, electrical engineering, etc. Therefore, it is important to have a mathematical apparatus that allows you to transform the graphical presentation of information about objects into algebraic models for their further research using purely mathematical methods. If necessary, it is always possible to return from such an algebraic model to a graphical representation of the object (for example, to a graphical representation of a circuit diagram in electronics). For each of the considered operations on graphs, either a combination of algebraic operations or an easily programmable matrix processing algorithm is proposed, which can be used to representanygraph. Attentionisalsopaidtothedifferencesinsuchprocessing depending on the type of graphs involved in the considered operations. Some operations are more convenient to perform with adjacency matrices, and some - with incidence matrices. This article also considers these features of matrix execution of operations on graphs. All the proposed algorithms are illustrated with specific detailed examples. Thus, it is shown that for all operations on graphically presented objects, their matrix interpretation is possible. This result greatly facilitates the possibility of software implementation of work with such graphic objects.
This article considers the possibility of matrix execution of both unary and binary operations on graphs. Graphs, as an abstract mathematical construction, have a very wide range of practical applications. First of all, it is algorithmization and computer processing of information, electrical engineering, etc. Therefore, it is important to have a mathematical apparatus that allows you to transform the graphical presentation of information about objects into algebraic models for their further research using purely mathematical methods. If necessary, it is always possible to return from such an algebraic model to a graphical representation of the object (for example, to a graphical representation of a circuit diagram in electronics). For each of the considered operations on graphs, either a combination of algebraic operations or an easily programmable matrix processing algorithm is proposed, which can be used to representanygraph. Attentionisalsopaidtothedifferencesinsuchprocessing depending on the type of graphs involved in the considered operations. Some operations are more convenient to perform with adjacency matrices, and some - with incidence matrices. This article also considers these features of matrix execution of operations on graphs. All the proposed algorithms are illustrated with specific detailed examples. Thus, it is shown that for all operations on graphically presented objects, their matrix interpretation is possible. This result greatly facilitates the possibility of software implementation of work with such graphic objects.
Опис
Ключові слова
орiєнтований та неорiєнтований граф, матриця сумiжностi, матриця iнцидентностi, операцiї над графами, елементарнi логiчнi операцiї, булева матриця, багатозначна логiка, directed and undirected graph, adjacency matrix, incidence matrix, operations on graphs, elementary logical operations, Boolean matrix, multivalued logic
Бібліографічний опис
Дослідження в математиці і механіці = Researches in mathematics and mechanics