Graph operations | Graph families | Bipartite graphs
In graph theory, a branch of mathematics, the simplex graph κ(G) of an undirected graph G is itself a graph, with one node for each clique (a set of mutually adjacent vertices) in G. Two nodes of κ(G) are linked by an edge whenever the corresponding two cliques differ in the presence or absence of a single vertex. The empty set is included as one of the cliques of G that are used to form the clique graph, as is every set of one vertex and every set of two adjacent vertices. Therefore, the simplex graph contains within it a subdivision of G itself. The simplex graph of a complete graph is a hypercube graph, and the simplex graph of a cycle graph of length four or more is a gear graph. The simplex graph of the complement graph of a path graph is a Fibonacci cube. The complete subgraphs of G can be given the structure of a median algebra: the median of three cliques A, B, and C is formed by the vertices that belong to a majority of the three cliques. Any two vertices belonging to this median set must both belong to at least one of A, B, or C, and therefore must be linked by an edge, so the median of three cliques is itself a clique. The simplex graph is the median graph corresponding to this median algebra structure. When G is the complement graph of a bipartite graph, the cliques of G can be given a stronger structure as a distributive lattice, and in this case the simplex graph is the graph of the lattice. As is true for median graphs more generally, every simplex graph is itself bipartite. The simplex graph has one vertex for every simplex in the clique complex X(G) of G, and two vertices are linked by an edge when one of the two corresponding simplexes is a facet of the other. Thus, the objects (vertices in the simplex graph, simplexes in X(G)) and relations between objects (edges in the simplex graph, inclusion relations between simplexes in X(G)) are in one-to-one correspondence between X(G) and κ(G). Simplex graphs were introduced by , who observed that a simplex graph has no cubes if and only if the underlying graph is triangle-free, and showed that the chromatic number of the underlying graph equals the minimum number n such that the simplex graph can be isometrically embedded into a Cartesian product of n trees. As a consequence of the existence of triangle-free graphs with high chromatic number, they showed that there exist two-dimensional topological median algebras that cannot be embedded into products of finitely many real trees. also use simplex graphs as part of their proof that testing whether a graph is triangle-free or whether it is a median graph may be performed equally quickly. (Wikipedia).
Graph of x^2 + 6xb + 5b^2 as b varies
From playlist 3d graphs
From playlist 3d graphs
Graph of x^2 + y^2 + pxy as p varies
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Graph of x^2 + 6xy + 5y^2 rotating
From playlist 3d graphs
From playlist 3d graphs
Graph Theory FAQs: 01. More General Graph Definition
In video 02: Definition of a Graph, we defined a (simple) graph as a set of vertices together with a set of edges where the edges are 2-subsets of the vertex set. Notice that this definition does not allow for multiple edges or loops. In general on this channel, we have been discussing o
From playlist Graph Theory FAQs
From playlist 3d graphs
Graphs in the Complex Plane (1 of 4: Introductory Examples)
More resources available at www.misterwootube.com
From playlist Complex Numbers
This video explains the definitions of simple graphs, multigraphs, connected and not connected graphs, complete graphs, and the Handshake lemma. mathispower4u.com
From playlist Graph Theory (Discrete Math)
A Fun Thing You Can Do With Topological Combinatorics
This video aims at explaining the connection between the Borsuk Ulam Theorem in topology and its connection with the Tucker's Lemma in the conbinatorics branch of math. The content is based on a direct reading program that took place from Feb. 2022 to Sep. 2022 in University of California,
From playlist Summer of Math Exposition 2 videos
Omer Bobrowski: Random Simplicial Complexes, Lecture I
A simplicial complex is a collection of vertices, edges, triangles, tetrahedra and higher dimensional simplexes glued together. In other words, it is a higher-dimensional generalization of a graph. In recent years there has been a growing effort in developing the theory of random simplicia
From playlist Workshop: High dimensional spatial random systems
Omer Bobrowski: Random Simplicial Complexes II
A simplicial complex is a collection of vertices, edges, triangles, tetrahedra and higher dimensional simplexes glued together. In other words, it is a higher-dimensional generalization of a graph. In recent years there has been a growing effort in developing the theory of random simplicia
From playlist Workshop: High dimensional spatial random systems
Jonathan Barmak: Star clusters in clique complexes and the Vietoris-Rips complex of planar sets
Abstract: The star cluster of a simplex in a simplicial complex K is the union of the stars of its vertices. When K is clique, star clusters are contractible. We will recall applications of this notion to the study of homotopy invariants of independence complexes of graphs. If X is a plan
From playlist Vietoris-Rips Seminar
Simplicial Complexes - Your Brain as Math Part 2 | Infinite Series
Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi What is a Simplicial Complex and how can it help us decode the brain’s neurological structure? This is Part 2 in our Your Brain as Math mini-series. Check out Part 1 h
From playlist Your Brain as Math
Francis BROWN - Graph Complexes, Invariant Differential Forms and Feynman integrals
Kontsevich introduced the graph complex GC2 in 1993 and raised the problem of determining its cohomology. This problem is of renewed importance following the recent work of Chan-Galatius-Payne, who related it to the cohomology of the moduli spaces Mg of curves of genus g. It is known by Wi
From playlist Algebraic Structures in Perturbative Quantum Field Theory: a conference in honour of Dirk Kreimer's 60th birthday
Topological spaces for directed graphs [Henri Riihimäki]
Directed graphs serve as a model for various phenomena in the sciences, for example networks of neurons in the brain or gene regulatory networks. To apply TDA tools we need to construct higher dimensional topological spaces out of directed graphs. In this tutorial we will learn about two s
From playlist Tutorial-a-thon 2021 Fall
Samir Shukla (3/31/23): Vietoris-Rips complexes of hypercube graphs
In this talk, we discuss the Vietoris-Rips complexes of hypercube graphs. These questions on hypercubes arose from work by Kevin Emmett, Raúl Rabadán, and Daniel Rosenbloom related to the persistent homology formed from genetic trees, reticulate evolution, and medial recombination. A (fin
From playlist Vietoris-Rips Seminar
Topological Message Passing on GNN | SIMPLICIAL COMPLEXES on CW Networks #ai
We go from Message Passing GNN (MPGNN) to TOPOLOGICAL Message Passing on CW Networks: Lifting a Graph to a higher topological space allows for high-dimensional interactions (greater than 2) given our higher-dim topological spaces. Computational Graph Neural Networks increase its complexiti
From playlist Learn Graph Neural Networks: code, examples and theory
V5-04. Linear Programming. Matrix representation of the Simplex Algorithm.
Math 484: Linear Programming. Matrix representation of the Simplex Algorithm. Wen Shen, 2020, Penn State University
From playlist Math484 Linear Programming Short Videos, summer 2020