Download Discrete math

Discrete math
Objectives: To learn vocabulary, vertex/edge patterns in discrete
math.
Vertex Edge Graph - A collection of points
some of which are joined by line segments
or curves.
This graph has 6 vertices and 7 edges
Each point is a vertex and each line is an edge
Degrees of a vertex
The degree of a vertex in a graph is
the number of edges that touch it.
3
2
4
3
2
3
Each vertex is labeled
with its degree
3
A graph is regular if every vertex has the same degree.
2
2
2
Euler Graph (pronounced oilier)
An Eulerian path in a graph is a path that travels
along every edge of the graph exactly once. An
Eulerian path might pass through individual
vertices of the graph more than once.
A Euler path is a
snowplow problem
where a snow plow
needs to plow every
street once.
Start and finish
Euler circuits is a path that ends at the same vertex it started
Draw the Vertex/edge graph and answer the following questions.
1) How many vertices are there?
6
2) How many edges are there?
9
3)How many vertices have a degree of 2? 3
4) How many vertices have a degree of 4? 3
Draw a Euler circuit starting at the vertex with a white dot.
Remember: A circuit travels along every path exactly once and
may pass through vertices multiple times before ending at the
starting vertex.
Use this chart to see if a Euler path or circuit may be drawn
# of ODD Vertices
Implication (for a connected graph)
0
There is at least
one Euler Circuit.
1
THIS IS IMPOSSIBLE! Can’t be drawn
2
There is no Euler Circuit
but at least 1 Euler Path.
more than 2
There are no Euler Circuits
or Euler Paths.
Vocabulary:
Hamiltonian Circuit/Paths:
A Hamiltonian path in a graph is a path that
passes through every vertex in the graph exactly
once. A Hamiltonian path does not necessarily pass
through all the edges of the graph, however.
A Hamiltonian path which ends in the same place in
which it began is called a Hamiltonian circuit.
Trace a Hamiltonian path
Only a path, not a circuit. The path did
not end at the same vertex it started.
The path does not need to go over every edge but it can only go
over an edge once and must pass through every vertex exactly
once.
Hamiltonian Circuits
are often called the
mail man circuit
because the mailman
goes to every mailbox
but does not need to go
over every street.
Video
Store
Home
Sweet
Home
Find the quickest
route from home
to the pharmacy
5
6
N
3
Bank
4
4
3
4
Dry
Cleaners
Grocery
Store
6
6
2
4
5
Pharmacy
Post Office
A weighted edge has a value assigned to the edge
(i.e.- miles, time, gallons of gas, any unit of measure)
Review:
Euler Graphs
Hamiltonian
Graphs
Circuits
Passes over edge exactly once. May pass
through a vertex more than once.
Passes through every vertex exactly once but
not necessarily over every edge.
The path ends at the same vertex it started.