1. Graphs - 11 CSC1001 Discrete Mathematics 1
CHAPTER
กราฟ
11 (Graphs)
1 Introduction to Graphs and Graph Models
1. Deffinition of Graphs
Definition 1
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.
Types of Graphs
1) A graph in which each edge connects two different vertices and where no two edges connect the same
pair of vertices is called a simple graph.
2) A graphs that may have multiple edges connecting the same vertices are called multigraphs.
3) A graphs that may include loops, and possibly multiple edges connecting the same pair of vertices or a
vertex to itself, are sometimes called pseudographs.
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2. 2 CSC1001 Discrete Mathematics 11 - Graphs
4) From types of graphs in type 1-3 are undirected graphs. However, to construct a graph model, we may
find it necessary to assign directions to the edges of a graph. This is called a directed graph (or digraph).
We directed graph
A directed graph which has no loops and has no multiple directed edges is called a simple directed
graph.
A directed graph that may have multiple directed edges from a vertex to a second (possibly the same)
vertex is called directed multigraphs.
5) A graph with both directed and undirected edges is called a mixed graph.
Example 1 (6 points) Identify the properties of graph terminology in the blank from the table.
Multiple Edges Loops
No. Type Edges
Allowed? Allowed?
1. Simple graph
2. Multigraph
3. Pseudograph
4. Simple directed graph
5. Directed multigraph
6. Mixed graph
2. Graph Models
Graphs are used in a wide variety of models. Can you find a subject to which graph theory has been or
has not been applied?
1) Social Networks Graphs are extensively used to model social structures based on different kinds of
relationships between people or groups of people. In these graph models, individuals or organizations are
represented by vertices; relationships between individuals or organizations are represented by edges.
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3. Graphs - 11 CSC1001 Discrete Mathematics 3
Acquaintanceship and Friendship Graphs
Influence Graphs
2) Communication Network We can model different communications networks using vertices to represent
devices and edges to represent the particular type of communications links of interest.
Call Graphs
3) Information Network Graphs can be used to model various networks that link particular types of infor-
mation. We can describe how to model the WorldWideWeb using a graph and we can also describe how
to use a graph to model the citations in different types of documents.
Web Graph WorldWideWeb can be modeled as a directed graph where each Web page is represented
by a vertex and where an edge starts at the Web page a and ends at the Web page b if there is a link
on a pointing to b.
Citation Graphs Graphs can be used to represent citations in different types of documents, including
academic papers, patents, and legal opinions. In such graphs, each document is represented by a
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4. 4 CSC1001 Discrete Mathematics 11 - Graphs
vertex, and there is an edge from one document to a second document if the first document cites the
second in its citation list.
4) Software Design Applications Graph models are useful tools in the design of software.
Module Dependency Graphs
Precedence Graphs
5) Transportation Network We can use graphs to model many different types of transportation networks,
including road, air, and rail networks, as well shipping networks
6) Biological Networks Many aspects of the biological sciences can be modeled using graphs.
Niche Overlap Graphs in Ecology
Protein Interaction Graphs
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5. Graphs - 11 CSC1001 Discrete Mathematics 5
7) Tournaments We now give some examples that show how graphs can also be used to model different
kinds of tournaments.
Round-Robin Tournaments
Single-EliminationTournaments
Example 2 (12 points) Determine whether the graph shown has directed or undirected edges, whether it has
multiple edges, and whether it has one or more loops. Use your answers from the type of graph in Example 1.
1) Graph No.1 2) Graph No.2
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6. 6 CSC1001 Discrete Mathematics 11 - Graphs
3) Graph No.3 5) Graph No.5
6) Graph No.6
4) Graph No.4
2 Graph Terminology and Special Types of Graphs
1. Deffinition of Graphs
Definition 1
Two 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 e of G. Such an edge e is called incident with the vertices u and v and e is said to
connect u and v.
Definition 2
The set of all neighbors of a vertex v of G = (V,E), denoted by N(v), is called the neighbourhood of v. If A is
a subset of V, we denote by N(A) the set of all vertices in G that are adjacent to at least one vertex in A.
So, N(A) = Uv∈A N(v) .
Definition 3
The degree of a vertex in an undirected graph is the number of edges incident with it, except that a loop at
a vertex contributes twice to the degree of that vertex. The degree of the vertex v is denoted by deg(v).
Definition 4
THE HANDSHAKING THEOREM Let G = (V ,E) be an undirected graph with m edges. Then
3m = ∑ deg( v) .
v∈V
(Note that this applies even if multiple edges and loops are present.)
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7. Graphs - 11 CSC1001 Discrete Mathematics 7
Example 3 (5 points) What are the degrees and what are the neighborhoods of the vertices in the graphs G
and H displayed in Figure?
Example 4 (2 points) How many edges are there in a graph with 10 vertices each of degree six?
Example 5 (2 points) How many edges are there in a graph with 12 vertices each of degree four?
Definition 5
When (u, v) is an edge of the graph G with directed edges, u is said to be adjacent to v and v is said to be
adjacent from u. The vertex u is called the initial vertex of (u, v), and v is called the terminal or end vertex of
(u, v). The initial vertex and terminal vertex of a loop are the same.
Definition 6
In a graph with directed edges the in-degree of a vertex v, denoted by deg−(v), is the number of edges with
v as their terminal vertex. The out-degree of v, denoted by deg+(v), is the number of edges with v as their
initial vertex. (Note that a loop at a vertex contributes 1 to both the in-degree and the out-degree of this
vertex.)
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