Graph Concepts: Difference between revisions
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=Graph Definition= | =Graph Definition= | ||
A '''graph''' is a pair-wise relationship among a set of objects. Mathematically, a graph G is a pair (V, E), where V is a finite set of [[#Vertex|vertices]], called the '''vertex set''' of G, and E is a [[Relations#Overview|binary relation]] on G, called the '''edge set''' of G, which contains the graph's [[#Edge|edges]]. | A '''graph''' is a pair-wise relationship among a set of objects. Mathematically, a graph G is a pair (V, E), where V is a finite set of [[#Vertex|vertices]], called the '''vertex set''' of G, and E is a [[Relations#Overview|binary relation]] on G, called the '''edge set''' of G, which contains the graph's [[#Edge|edges]]. | ||
==<tt>n</tt> and <tt>m</tt> Convention== | |||
In graph problems, the usual convention is to denote the number of vertices with n and the number of edges with m. In most, but not all, applications, m is Ω(n) and O(n<sup>2</sup>). Also see [[#Sparse_Graphs|sparse graphs]] and [[#Dense_Graphs|dense graphs]]. | |||
=<span id='Vertex'></span><span id='Node'></span><span id='Vertices_and_Edges'></span>Vertex (Node)= | =<span id='Vertex'></span><span id='Node'></span><span id='Vertices_and_Edges'></span>Vertex (Node)= | ||
Revision as of 19:51, 1 October 2021
Internal
Graph Definition
A graph is a pair-wise relationship among a set of objects. Mathematically, a graph G is a pair (V, E), where V is a finite set of vertices, called the vertex set of G, and E is a binary relation on G, called the edge set of G, which contains the graph's edges.
n and m Convention
In graph problems, the usual convention is to denote the number of vertices with n and the number of edges with m. In most, but not all, applications, m is Ω(n) and O(n2). Also see sparse graphs and dense graphs.
Vertex (Node)
An elements of the vertex set V is called vertex (plural vertices). An alternate term for vertex, used sometimes in the graph theory literature, is node. We prefer to use the term "node" when we refer to the vertices of rooted trees. We use "vertex" as a more generic term that refers to graphs in general. Another alternate name is entity.
Edge (Arc)
The element of the edge set E are called edges (also known as relationships or arcs). By convention, we use (u, v) notation for an edge, where u and v represent vertices in V. The order in which vertices are specified for an edge may be relevant. If the order in which the vertices are specified matters, then the graph is a directed graph. If the order in which the vertices are specified does not matter, then the graph is an undirected graph.