Graph invariants | Geometric intersection | Topological graph theory
In graph theory, the crossing number cr(G) of a graph G is the lowest number of edge crossings of a plane drawing of the graph G. For instance, a graph is planar if and only if its crossing number is zero. Determining the crossing number continues to be of great importance in graph drawing, as user studies have shown that drawing graphs with few crossings makes it easier for people to understand the drawing. The study of crossing numbers originated in Turán's brick factory problem, in which Pál Turán asked for a factory plan that minimized the number of crossings between tracks connecting brick kilns to storage sites. Mathematically, this problem can be formalized as asking for the crossing number of a complete bipartite graph. The same problem arose independently in sociology at approximately the same time, in connection with the construction of sociograms. Turán's conjectured formula for the crossing numbers of complete bipartite graphs remains unproven, as does an analogous formula for the complete graphs. The crossing number inequality states that, for graphs where the number e of edges is sufficiently larger than the number n of vertices, the crossing number is at least proportional to e3/n2. It has applications in VLSI design and incidence geometry. Without further qualification, the crossing number allows drawings in which the edges may be represented by arbitrary curves. A variation of this concept, the rectilinear crossing number, requires all edges to be straight line segments, and may differ from the crossing number. In particular, the rectilinear crossing number of a complete graph is essentially the same as the minimum number of convex quadrilaterals determined by a set of n points in general position. The problem of determining this number is closely related to the happy ending problem. (Wikipedia).
What are Bridges of Graphs? | Graph Theory, Edge Deletion
What are bridges of graphs? Bridges are the edge version of cut vertices. If e is an edge of a graph G and deleting e disconnected the component it belongs to, then e is an edge. So, for a connected graph G, an edge e is a bridge of G if G-e is disconnected. For a disconnected graph G, e
From playlist Graph Theory
From playlist M. Graph Theory
Graph Theory: 34. Bridge edges
I define what a bridge edge is in a graph and provide several examples. Then I explain a proof that an edge is a bridge in a graph if and only if the edge is not in any cycle of the graph. An introduction to Graph Theory by Dr. Sarada Herke. Related Videos: http://youtu.be/SFFEc8DbO0Y -
From playlist Graph Theory part-7
Graph simple inequalities on a number line.
From playlist Algebra: Linear Equations with One Variable
What is a Path Graph? | Graph Theory
What is a path graph? We have previously discussed paths as being ways of moving through graphs without repeating vertices or edges, but today we can also talk about paths as being graphs themselves, and that is the topic of today's math lesson! A path graph is a graph whose vertices can
From playlist Graph Theory
The Definition of a Graph (Graph Theory)
The Definition of a Graph (Graph Theory) mathispower4u.com
From playlist Graph Theory (Discrete Math)
What is a Graph? | Graph Theory
What is a graph? A graph theory graph, in particular, is the subject of discussion today. In graph theory, a graph is an ordered pair consisting of a vertex set, then an edge set. Graphs are often represented as diagrams, with dots representing vertices, and lines representing edges. Each
From playlist Graph Theory
Graph Theory 37. Which Graphs are Trees
A proof that a graph of order n is a tree if and only if it is has no cycle and has n-1 edges. An introduction to Graph Theory by Dr. Sarada Herke. Related Videos: http://youtu.be/QFQlxtz7f6g - Graph Theory: 36. Definition of a Tree http://youtu.be/Yon2ndGQU5s - Graph Theory: 38. Three
From playlist Graph Theory part-7
Knots, Virtual Knots and Virtual Knot Cobordism by Louis H. Kauffman
PROGRAM KNOTS THROUGH WEB (ONLINE) ORGANIZERS: Rama Mishra, Madeti Prabhakar, and Mahender Singh DATE & TIME: 24 August 2020 to 28 August 2020 VENUE: Online Due to the ongoing COVID-19 pandemic, the original program has been canceled. However, the meeting will be conducted through onl
From playlist Knots Through Web (Online)
8ECM Public Lecture: Bojan Mohar
From playlist 8ECM Public Lectures
26. Sum-product problem and incidence geometry
MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019 Instructor: Yufei Zhao View the complete course: https://ocw.mit.edu/18-217F19 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP62qauV_CpT1zKaGG_Vj5igX A famous open problem says that no set of integers can si
From playlist MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019
15. Graph limits II: regularity and counting
MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019 Instructor: Yufei Zhao View the complete course: https://ocw.mit.edu/18-217F19 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP62qauV_CpT1zKaGG_Vj5igX Prof. Zhao explains how graph limits can be used to gener
From playlist MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019
Accidental Crossings - Graph Theory
Example and justification for accidental crossings in graph theory
From playlist Graph Theory
Graph Theory: 01. Seven Bridges of Konigsberg
The Seven Bridges of Konigsberg Problem was solved by Euler in 1735 and that was the beginning of Graph Theory! In this video, we explain the problem and the method that Euler used to solve it. In this introductory video, no previous knowledge of Graph Theory will be assumed. --An intro
From playlist Graph Theory part-1
Which Complete Graphs are Planar? | Graph Theory
Which complete graphs are planar? Which complete graphs are nonplanar? We'll answer this question in today's graph theory lesson! We'll see that K1, K2, K3, and K4 are all planar complete graphs. Then, we'll prove that K5 is nonplanar and see why that implies no complete graph with at le
From playlist Graph Theory
What are Planar Graphs? | Graph Theory
What are planar graphs? How can we draw them in the plane? In today's graph theory lesson we'll be defining planar graphs, plane graphs, regions of plane graphs, boundaries of regions of plane graphs, and introducing Euler's formula for connected plane graphs. A planar graph is a graph t
From playlist Graph Theory
A conversation between Louis Kauffman and Stephen Wolfram at the Wolfram Summer School 2021
Stephen Wolfram plays the role of Salonnière in this new, on-going series of intellectual explorations with special guests. Watch all of the conversations here: https://wolfr.am/youtube-sw-conversations Follow us on our official social media channels. Twitter: https://twitter.com/Wolfra
From playlist Conversations with Special Guests
Proof: Euler's Formula for Plane Graphs | Graph Theory
We'll be proving Euler's theorem for connected plane graphs in today's graph theory lesson! Commonly know by the equation v-e+f=2, or in more common graph theory notation n-m+r=2, we'll prove this famous result using a minimum counterexample proof! The result states that, for connected pl
From playlist Graph Theory
Due to the COVID-19 pandemic, Carnegie Mellon University is protecting the health and safety of its community by holding all large classes online. People from outside Carnegie Mellon University are welcome to tune in to see how the class is taught, but unfortunately Prof. Loh will not be o
From playlist CMU 21-228 Discrete Mathematics
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