Matrix representation of operations on graphs

Abstract

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 represent any graph. Attention is also paid to the differences in such processing 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.