Download Graph Section 15.1

Chapter 15
Graph
Theory
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Chapter 15: Graph Theory
15.1
15.2
15.3
15.4
Basic Concepts
Euler Circuits
Hamilton Circuits and Algorithms
Trees and Minimum Spanning Trees
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Chapter 1
Section 15-1
Basic Concepts
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Basic Concepts
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•
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Graphs
Walks, Paths, and Circuits
Complete Graphs and Subgraphs
Graph Coloring
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Simple Graphs
Graphs in which there is no more than one edge
between any two vertices and in which no edge goes
from vertex to the same vertex are called simple.
Simple
Not simple
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Degree of the Vertex
The number of edges joined to a vertex is called the
degree of the vertex.
A
C
D
B
A has degree 0, B has degree 1, C has degree 2
and D has degree 3.
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Isomorphic Graphs
Two graphs are isomorphic if there is a one-to-one
matching between vertices of the two graphs with
the property that whenever there is an edge
between two vertices of either one of the graphs,
there is an edge between the corresponding vertices
of the other graph.
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Connected and Disconnected Graphs
A graph is connected if one can move from each
vertex of the graph to every other vertex of the graph
along edges of the graph. If not, the graph is
disconnected. The connected pieces of a graph are
called the components of the graph.
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Example: Isomorphic Graphs
Are the two graphs isomorphic?
Solution
Yes, and corresponding vertices are labeled
below.
A
D
B
C
B
A
D
C
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Sum of Degrees Theorem
In any graph, the sum of the degrees of the vertices
equals twice the number of edges.
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Example: Sum of Degrees
A graph has exactly four vertices, each of degree 3.
How many edges does this graph have?
Solution
The sum of degrees of the vertices is 12. By the
theorem, this number is twice the number of edges,
so the number of edges is 12/2 = 6.
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Walk
A walk in a graph is a sequence of vertices, each
linked to the next vertex by a specified edge of the
graph.
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Path
A path in a graph is a walk that uses no edge more
than once.
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Circuit
A circuit in a graph is a path that begins and ends at
the same vertex.
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Venn Diagram of Walks, Paths, and
Circuits
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Example: Classifying Walks
Using the graph, classify each sequence as a walk, a
path or a circuit.
A
E
B
a) E  C  D  E
D
C
b) A  C  D  E  B  A
c) B  D  E  B  C
d) A  B  C  D  B  A
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Example: Classifying Walks
A
Solution
E
B
D
C
Walk
Path
Circuit
a) No
No
No
b) Yes
Yes
Yes
c) Yes
No
No
d) Yes
No
No
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Complete Graph
A complete graph is a graph in which there is
exactly one edge going from each vertex to each
other vertex in the graph.
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Example: Complete Graph
Draw a complete graph with four vertices.
Solution
Answers may vary, but one solution is shown
below.
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Subgraph
A graph consisting of some of the vertices of the
original graph and some the original edges between
those vertices is called a subgraph.
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Coloring and Chromatic Number
A coloring for a graph is a coloring of the vertices in
such a way that the vertices joined by an edge have
different colors. The chromatic number of a graph
is the least number of colors needed to make a
coloring.
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Coloring a Graph
Step 1:
Choose a vertex with highest degree, and
color it. Use the same color to color as
many vertices as you can without coloring
vertices joined by an edge of the same
color.
Step 2:
Choose a new color, and repeat what you
did in Step 1 for vertices not already
colored.
Step 3:
Repeat Step 1 until all vertices are colored.
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Example: Coloring a Graph
Color the graph below and give its chromatic
number .
Solution
Its chromatic
number is 3.
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