In graph theory, an adjacent vertex of a vertex v in a graph is a vertex that is connected to v by an edge. The neighbourhood of a vertex v in a graph G is the subgraph of G induced by all vertices adjacent to v, i.e., the graph composed of the vertices adjacent to v and all edges connecting vertices adjacent to v. The neighbourhood is often denoted or (when the graph is unambiguous) . The same neighbourhood notation may also be used to refer to sets of adjacent vertices rather than the corresponding induced subgraphs. The neighbourhood described above does not include v itself, and is more specifically the open neighbourhood of v; it is also possible to define a neighbourhood in which v itself is included, called the closed neighbourhood and denoted by . When stated without any qualification, a neighbourhood is assumed to be open. Neighbourhoods may be used to represent graphs in computer algorithms, via the adjacency list and adjacency matrix representations. Neighbourhoods are also used in the clustering coefficient of a graph, which is a measure of the average density of its neighbourhoods. In addition, many important classes of graphs may be defined by properties of their neighbourhoods, or by symmetries that relate neighbourhoods to each other. An isolated vertex has no adjacent vertices. The degree of a vertex is equal to the number of adjacent vertices. A special case is a loop that connects a vertex to itself; if such an edge exists, the vertex belongs to its own neighbourhood. (Wikipedia).
Graph Theory: 11. Neighbourhood and Bipartite Test with Colours
In this video I provide the definition of the neighbourhood of a vertex and then describe a colouring algorithm that uses the neighbourhoods of vertices in order to determine whether or not a given graph is bipartite. An introduction to Graph Theory by Dr. Sarada Herke.
From playlist Graph Theory part-2
Neighborhood of a Vertex | Open and Closed Neighborhoods, Graph Theory
What is the neighborhood of a vertex? Remember that the neighbors of a vertex are its adjacent vertices. So what do you think its neighborhood is? We’ll be going over neighborhoods, both open neighborhoods and closed neighborhoods, and an alternative definition of neighborhood, in today’s
From playlist Graph Theory
Graph Theory: 05. Connected and Regular Graphs
We give the definition of a connected graph and give examples of connected and disconnected graphs. We also discuss the concepts of the neighbourhood of a vertex and the degree of a vertex. This allows us to define a regular graph, and we give some examples of these. --An introduction to
From playlist Graph Theory part-1
In-Neighborhoods and Out-Neighborhoods in Digraphs | Graph Theory
We discuss neighborhoods in the context of directed graphs. This requires that we split the concept of "neighborhood" in two, since a vertex v could be adjacent TO a vertex u, or adjacent FROM the vertex u. Thus, the outneighborhood of a vertex v is the set of vertices v is adjacent to, an
From playlist Graph Theory
What are Connected Graphs? | Graph Theory
What is a connected graph in graph theory? That is the subject of today's math lesson! A connected graph is a graph in which every pair of vertices is connected, which means there exists a path in the graph with those vertices as endpoints. We can think of it this way: if, by traveling acr
From playlist Graph Theory
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
In this tutorial I explore the concepts of walks, trails, paths, cycles, and the connected graph.
From playlist Introducing graph theory
Graph Theory: 02. Definition of a Graph
In this video we formally define what a graph is in Graph Theory and explain the concept with an example. In this introductory video, no previous knowledge of Graph Theory will be assumed. --An introduction to Graph Theory by Dr. Sarada Herke. This video is a remake of the "02. Definitio
From playlist Graph Theory part-1
Graph Theory: 51. Eccentricity, Radius & Diameter
Eccentricity, radius and diameter are terms that are used often in graph theory. They are related to the concept of the distance between vertices. The distance between a pair of vertices is the length of a shortest path between them. We begin by reviewing some of the properties of dista
From playlist Graph Theory part-9
What is the limit of a sequence of graphs?? | Benjamini-Schramm Convergence
This is an introduction to the mathematical concept of Benjamini-Schramm convergence, which is a type of graph limit theory which works well for sparse graphs. We hope that most of it is understandable by a wide audience with some mathematical background (including some prior exposure to g
From playlist Summer of Math Exposition Youtube Videos
Geometry of Surfaces - Topological Surfaces Lecture 2 : Oxford Mathematics 3rd Year Student Lecture
This is the second of four lectures from Dominic Joyce's 3rd Year Geometry of Surfaces course. The four lectures cover topological surfaces and conclude with a big result, namely the classification of surfaces. This lectures covers building topological surfaces by gluing sides of polygons.
From playlist Oxford Mathematics Student Lectures - Geometry of Surfaces
Clelia Pech: Curve neighbourhoods for odd symplectic Grassmannians
CIRM VIRTUAL CONFERENCE Odd symplectic Grassmannians are a family of quasi-homogeneous varieties with properties nevertheless similar to those of homogeneous spaces, such as the existence of a Schubert-type cohomology basis. In this talk based on joint work with Ryan Shifler, I will expl
From playlist Virtual Conference
How do you get a society that provides basic decent services to all citizens? Political theorist John Rawls had a good idea, and it was called 'the veil of ignorance.' SUBSCRIBE to our channel for new films every week: http://tinyurl.com/o28mut7 If you like our films take a look at our sho
From playlist GREAT IDEAS
Proof: Hall's Marriage Theorem for Bipartite Matchings | Graph Theory
A bipartite graph G with partite sets U and W, where |U| is less than or equal to |W|, contains a matching of cardinality |U|, as in, a matching that covers U, if and only if for every subset S of U, |S| is less than or equal to the cardinality of the neighborhood of S (as in - S has as ma
From playlist Graph Theory
Andrea D'Agnolo : On the Riemann-Hilbert correspondence for irregular holonomic D-modules
Abstract: The classical Riemann-Hilbert correspondence establishes an equivalence between the triangulated categories of regular holonomic D-modules and of constructible sheaves. In a joint work with Masaki Kashiwara, we proved a Riemann-Hilbert correspondence for holonomic D-modules which
From playlist Analysis and its Applications
Embedding Graphs with Deep Learning
This video explains how to Embed Graphs with Deep Learning. This includes showing the difference between Matrix Decomposition and Deep learning methods as well. Thanks for watching! www.henryailabs.com
From playlist Deep Learning on 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
Wanlin Li - The Ceresa class: tropical, topological, and local - AGONIZE mini-conference
The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve, which is trivial in the Chow ring when the curve is hyperelliptic. Its image under a certain cycle class map provides a class in étale cohomology called the Ceresa class. There are few examples where the Ceresa cl
From playlist Arithmetic Geometry is ONline In Zoom, Everyone (AGONIZE)