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Graphs, Hypergraphs and Minors of their Incidence Matrices

Student: Safronov Denis

Supervisor: Dmitry Gribanov

Faculty: Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod)

Educational Programme: Applied Mathematics and Information Science (Bachelor)

Final Grade: 8

Year of Graduation: 2019

As is known, one of the priority departments of the modern optimization theory is integer linear programming and combinatorial optimization. Tasks of this kind often represent issues of maximizing or minimizing a definite linear functional on a certain polyhedron defined by a system of linear inequalities. Among the range of challenges under consideration, we can identify a subclass of problems whose polyhedrons are defined by systems of linear inequalities with a limited spectrum of minors. In the article by J. W. Grossman «On the Minors of an Incidence Matrix and Its Smith Normal Form» the interrelation between the spectrum of the minors for incidence matrices of simple graphs and the structure of these graphs are considered. An incidence matrix is a form of representation of the graph, in which the connections between edges and vertices are indicated. The columns of the incidence matrix correspond to the edges, while the rows related to the vertices. A non-zero value in the matrix cell indicates the incidence between corresponding edges and vertices. The outputs of this article help to better understand the structure of matrices and systems with a limited spectrum of minors. A hypergraph is a generalization of an ordinary graph whose edges can contain not only two elements, but also any subsets of vertices. Such objects have long been known in mathematics, but the term “hypergraph” was introduced due to the prosperous distribution of many concepts and methods of the theory of ordinary graphs to them. The purpose of the research is to formulate and describe the minor characteristics of the incidence matrices of some hypergraph classes. The study is relevant because it finds some applications in the theory of issues of integer linear programming, which is a significant part of modern optimization theory. Specific attention will be paid to the classes of Unicyclic Hypergraphs, Hypergraphs generated by Circulants of a distinct type and Hypergraphs with the number of equal 1 packing cycles; particularly to their minimal and maximal minors. The main target is to try applying the methods and results studied in the past two works to a wider class of objectives.

Full text (added May 22, 2019)

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