Vital Sine

This video explains walks, trails, and paths in hypergraphs, including plenty of examples to demonstrate these concepts. The focus of the video is on how these concepts generalize from graphs to hypergraphs, and how the generalizations are different from their origins in graph theory. We also cover the intuition about the different "types of movements" that each type of walk represents in a hypergraph. Resources- Books: "Introduction to Graph and Hypergraph Theory" by Vitaly I. Voloshin https://link.springer.com/book/10.1007/978-3-319-00080-0 Papers: https://arxiv.org/pdf/1608.01040.pdf https://arxiv.org/pdf/2006.16377.pdf https://arxiv.org/abs/1911.06523

This video introduces projections and layers of graph products, with plenty of visual examples. We will go over some of the properties of projections and layers as well. A projection is a function from the vertex set of a graph product to the vertex set of one of its factors, while a layer is a set of vertices in a graph product with certain coordinates held fixed and a single coordinate allowed to vary.

This video introduces clique complexes, a type of abstract simplicial complex. We will cover several examples and special cases of clique complexes. We'll go over the relationship between clique complexes and conformal hypergraphs, as well as certain requirements that clique complexes must satisfy. This video will also introduce independence complexes. Both independence complexes and clique complexes are abstract simplicial complexes that are built from a simple graph.

We introduce abstract simplicial complexes and the most important definitions, basic concepts, and terminology relating to them. We will use plenty of examples to demonstrate. We also look at how abstract simplicial complexes are a special type of hypergraph and look at the basic concepts relating to abstract simplicial complexes in the context of hypergraphs.

This video covers the graph operations known as p-sum, extended p-sum, and NEPS (non-extended p-sum). Each of these operations takes an ordered n-tuple of graphs as input, and outputs a graph with vertex set equal to the cartesian product of the input graph's vertex sets. However, these operations differ in how their edge sets are defined. The NEPS is the most general of these operations and contains the other two as special cases. The p-sum is the least general of these operations and is a special case of both the NEPS and the extended p-sum. These operations can be used to represent certain graph products (cartesian, tensor, strong products), but are more general than graph products as well. Links for more information: https://www.jstor.org/stable/43667432

This video introduces line graphs and 2-sections of hypergraphs, using several examples to demonstrate each concept. We will practice finding line graphs and 2-sections together, as well as look at some basic theorems and properties of line graphs and 2-sections. Thanks for watching!

This video introduces hypergraph parameters, including cover number, matching number, strong independence number, independence number, and transversal number, as well as their corresponding sets (coverings, matchings, strong independent sets, independent sets, and transversals/vertex covers). We look at several relationships between these parameters, and relate the parameters to the dual operation. The video includes many examples of each parameter and its corresponding set of vertices or edges in a hypergraph. Thanks for watching! Resources: https://link.springer.com/book/10.1007/978-3-319-00080-0 "Introduction to Graph and Hypergraph Theory" by Vitaly I. Voloshin

This video introduces hypergraph operations with examples. We look at the following operations: 1. weak vertex deletion 2. weak edge deletion 3. strong vertex deletion 4. strong edge deletion 5. edge contraction. We look at the relationship between these operations and the incidence matrix, and incidence graph, of hypergraphs, as well as what these operations mean in terms of the vertex and edge sets of a hypergraph. These operations will be useful for anyone interesting in studying hypergraphs more deeply.

This video covers collinearity graphs, a graph representation of an incidence structure, with several examples. We cover the basic properties of collinearity graphs, as well as how they can be used to define notions of connectivity and distance for incidence structures. We work through several examples of both finding collinearity graphs and applying them to find distances in an incidence structure, and then compare the notion of distance based upon collinearity graphs with that based upon incidence graphs, which we covered in my previous video. If you're interested in learning more, here are two great resources: https://link.springer.com/book/10.1007/978-3-319-43811-5 https://link.springer.com/book/10.1007/978-3-642-20972-7

This video covers incidence graphs, a graph representation of an incidence structure, with several examples. The incidence graph of an incidence structure coincides with the incidence graph of that structure when viewed as a hypergraph. We cover some of the basic properties of incidence graphs of incidence structures, and practice finding the incidence graphs of several incidence structures during the video. Finally, we discuss how incidence graphs can be used to define a notion of distance between objects in an incidence structure. For more information, see these resources: https://link.springer.com/book/10.1007/978-3-319-43811-5 https://link.springer.com/book/10.1007/978-3-642-20972-7

This video introduces incidence geometry, the study of incidence structures, with many examples. We cover incidence structures in their full generality, as well as contrast incidence geometry with other kinds of geometry. We discuss the matrix representations and pictorial representations of incidence structures, duality, and the relationship between incidence structures and graphs, specifically hypergraphs. Incidence geometry is a very rich subject, and if you would like to dive into this topic on your own, here are some excellent resources: https://link.springer.com/book/10.1007/978-3-319-43811-5 https://link.springer.com/book/10.1007/978-3-642-20972-7

This video covers incidence graphs, a concept from hypergraph theory, with many examples. We go over the basic properties of incidence graphs, as well as discuss how they relate to the idea of dual hypergraphs. An incidence graph is a bipartite graph representation of a hypergraph. If you are interested in learning more, here are some resources: BOOK: https://link.springer.com/book/10.1007/978-3-319-00080-0 BOOK: "Introduction to Graph and Hypergraph Theory" by Vitaly I. Voloshin Thanks for watching!