Section 15.1
Graphs, Paths, and Circuits
Objectives
1. Understand relationships in a graph.
2. Model relationships using graphs.
3. Understand and use the vocabulary of graph theory.
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Graphs
Graphs consist of:
1. Vertices, a finite set of points (singular is vertex), and
2. Line segments or curves, called edges.
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Graphs
Graphs consist of:
• An edge that starts and ends at the same vertex is called a loop.
• A graph consists of vertices and edges that start and end at
vertices.
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Graphs
Understanding Relationships in Graphs
Example: Explain why the figures below show equivalent
graphs.
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Graphs
Understanding Relationships in Graphs
Example: Explain why the figures below show equivalent
graphs.
Solution: In both figures, the vertices are A, B, C, and D. Both graphs
have an edge that connects vertex A to vertex B, an edge that
connects vertex B to vertex C, and an edge that connects vertex C
to vertex D. Because the two graphs have the same number of
vertices connected to each other in the same way, they are
equivalent.
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Modeling Relationships Using Graphs
We can illustrate a number of different situations that we
can represent, or model, using graphs.
Example: In the early 1700’s, the city of Königsberg,
Germany, was located on both banks and two islands of
the Pregel River. The figure below shows that the town’s
sections were connected by seven bridges.
Draw a graph that models the layout of Königsberg. Use
vertices to represent the land masses and edges to
represent the bridges.
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Modeling Relationships Using Graphs
Example Continued
Solution: The only thing that matters is the relationship
between land masses and bridges.
We label each of
the four land
masses with an
uppercase letter.
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We use points to
represent the land
masses. This figure
shows vertices A, B,
L, and R.
Section 15.1
We draw two edges
connecting vertex A
to right bank R, one
edge from A to B,
and two edges from
A to left bank L.
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Modeling Bordering Relationships
Example: The map of New England states are given below.
Draw a graph that models which New England states
share a common border. Use vertices to represent the
states and edges to represent common borders.
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Modeling Bordering Relationships
Example Continued
Solution:
a) Notice the states are labeled with their abbreviations.
b) We use the abbreviations to label each vertex and use
points to represent these vertices.
c) Whenever two states share a common border, we
connect the respective vertices with an edge.
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Circuits & Paths
Vocabulary of Graph Theory
• The degree of a vertex is the number of edges at that
vertex.
• A vertex with an even number of edges attached to it is
an even vertex.
• A vertex with an odd number of edges attached to it is an
odd vertex.
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Circuits & Paths
Vocabulary of Graph Theory
• Two vertices in a graph are said
to be adjacent vertices if there
is at least one edge connecting
them.
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Circuits & Paths
Identifying Adjacent Vertices
Example: List the pairs of adjacent vertices for the graph
given below.
Solution: A systematic way to
approach this problem is to
first list all the pairs of adjacent
vertices involving vertex A, then
all those involving vertex B but not A,
and so on. The adjacent vertices are A and B, A and C,
A and D, A and E, B and C, and E and E because there
is at least one edge connecting them.
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Circuits & Paths
Vocabulary of Graph Theory
• A path in a graph is a sequence of adjacent vertices and
the edges connecting them.
Notice the path along this graph is represented by
sequential arrows.
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Circuits & Paths
Vocabulary of Graph Theory
• A circuit is a path that begins and ends at the same
vertex.
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Circuits & Paths
Vocabulary of Graph Theory
• A graph is connected if for any two of its vertices there
is at least one path connecting them.
• A graph that is not connected is called disconnected.
• A disconnected graph is made of pieces that are
connected by themselves called components of the
graph.
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HW #4 - 48 every 4
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