Graphs
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Graphs A Graph G = (V, E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each edge has either one or two vertices associated with it, called its endpoints. An edge is said to connect its endpoints. A very simple Computer Network : An example of a simple graph It is also an undirected graph (edges have no Tampa Chicago Miami Denver LA No two different edge connect the same pair of vertices
description
Denver. Chicago. LA. Miami. Tampa. Graphs. A Graph G = (V, E) consists of V, a nonempty set of vertices (or nodes ) and E, a set of edges . Each edge has either one or two vertices associated with it, called its endpoints. An edge is said to connect its endpoints. - PowerPoint PPT Presentation
Transcript of Graphs
GraphsA Graph G = (V, E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each edge has either one or two vertices associated with it, called its endpoints. An edge is said to connect its endpoints.
A very simple Computer Network : An example of a simple graph
It is also an undirected graph (edges have no direction).
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No two different edge connect the same pair of vertices
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Now each edge has a direction associated to it. The edges are called directed edges and the graph is called a directed graph.
A directed edge is associated with an ordered pair of vertices (u, v). The edge is said to start at u and end at v.
Note that e = (u, v) and f = (v, u) are two different directed edges
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There are multiple edges (edges connecting the same pair of vertices).
The graph is not simple and is called (undirected) multigraphs.
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There are loops in the graph, and the graph is sometimes called a pseudograph.
Example: Acquaintanceship Graph
Use graph to represent various relationships between people.
John
Mary
Helen
Charles
Doug
Edge connecting two people when they know each other.
The graph is undirected with no multiple edges, or loops.
Example: Call Graphs
Use graph to model telephone calls made in a network (say a long distance telephone network).
352-343-3424
352-343-1453
352-343-6745
352-343-2563
Each directed edge (u, v) represents a call from u to v.
Vertices are phone numbers.
A directed graph
Example: The Web Graph
The World Wide Web can be modeled as a directed graph.
A vertex represents a web page and a directed edge (u, v) represents a link on u pointing to v.
WebCT
Cot3100Homepage
Slides
Extra Credit Assignment
Each directed edge (u, v) represents a link on u pointing to v.
Vertices are web pages.
A directed graph
Graph TerminologyTwo vertices u and v in an undirected graph G are called adjacent (or neighbors) in G if u and v are endpoints of an edge of G.
If e is associated with { u, v }, the edge e is called incident with the vertices u and v. The edge e is also said to connect u and v, and u and v are endpoints of e.
The degree of a vertex in an undirected graph is the number of edges incident with it (a loop contributes twice to the degree of that vertex).
John
Mary
Helen
Charles
Doug
deg ( Mary) = 3, deg(John) = 1, deg(Charles)=2, deg(Doug)= 2 and
deg(Helen) = 2.
The Handshaking TheoremLet G = (V, E) be an undirected graph with e edges. Then
2 e = Σv deg(v)
If we add all the degrees (of vertices), each edge will be counted twice. Hence the result.
Corollary: An undirected graph has an even number of vertices of odd degree.
2 e = Σv deg(v) = Σ v:odd deg(v) + Σ v:even deg(v)
John
Mary
Helen
Charles
Doug
deg ( Mary) = 3, deg(John) = 1, deg(Charles)=2, deg(Doug)= 2 and
deg(Helen) = 2.
The sum of all degrees is 10
There are 5 edges.
There are two vertices with odd degree.
352-343-3424 out-degree=1, in-dgree=2
352-343-1453In-degree=3out-degree=2
352-343-6745 out-degree=4
352-343-2563 in-degree = 3, out-degree=1
Cycles n vertices, v1, v2, ….., vn and edges
{ v1, v2 }, …, {vn-1, vn }
e
f
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a bc
d
{a, b, d }, {c, e, f, g}
K3, 4
Males, or warehouses
females, or suppliers
If a is in V1, then, b, d, e must be in V2 (why?)
Then, c is in V1 and there is no inconsistency.
So we can rearrange the graph as follows:
