§5.2 - 5.3 Graphs and
Graph Terminology
“Liesez Euler, Liesez Euler, c’est
notre maître à tous.”
- Pierre Laplace
Graphs consist of
 points called vertices
 lines called edges
1. Edges connect two
vertices.
2. Edges only intersect
at vertices.
3. Edges joining a
vertex to itself are
called loops.
Example 1: The following
picture
is a graph. List its vertices and
edges.
A
D
C
E
B
Example 2:
This is also a graph. The vertices just happen to have
people’s names.
Such a graph could represent friendships (or any kind of
relationship).
Flexo
Bender
Leela
Zoidberg
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (L ZW) d eco mpres sor
are nee ded to s ee this picture.
Fry
Amy
Farnsworth
Now check out the graph below.
What can we say about it in comparison to the previous
figure?
QuickTime™ and a
TIFF (L ZW) d eco mpres sor
are nee ded to s ee this picture.
Leela
Fry
Flexo
Amy
Bender
Farnsworth
Zoidberg
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Moral of the Story
• One graph may be drawn in (infinitely)
many ways, but it always provides us
with the same information.
• Graphs are a structure for describing
relationships between objects.
(The vertices denote the objects and the
edges represent the relationship.)
Graph Terminology
Graph Terminology
Graph Terminology
Example 3:
1) Find the degree of
each vertex.
A
D
2) Is A adjacent to B?
Is D adjacent to A?
Is E adjacent to itself?
Is C adjacent to
itself?
C
3) Is AB adjacent to BC?
Is CE adjacent to BD?
E
B
Graph Terminology
Graph Terminology
Graph Terminology
Example 4:
1) Find a path from B to K
passing through W but not
S.
J
S
B
W
H
K
2) Find a path from H to J of
length 4.
3) Find a circuit of length 5.
4) Find a circuit of length 1.
5) Find a bridge.
Example 5: Draw a picture of a graph that satisfies the following:
Vertices: A, B, C, D
Edges: AB, AC, AD, B is adjacent to D.
Graph Terminology
Example 6: The graph on the left has no
Euler paths, but the one on the right has
several.
R
R
D
A
L
D
A
L
§5.4 - 5.5 Graph Models and
Euler’s Theorems
“Now I will have less
distraction.”
- Leonhard Euler
after losing sight
in his right eye.
Königsberg’s Bridges II
(The rare sequel that is not entirely gratuitous.)
Recall from Tuesday the
puzzle that the
residents of Königsburg
had been unable to
solve until Euler’s
arrival:
• Is there a way to cross
all seven bridges
exactly once and return
to your starting point?
• Is there even a way to
cross all seven bridges
exactly once?
R
A
D
L
A stylized (i.e. - inaccurate)
map of Königsberg’s Bridges.
What Euler realized was that most of the information
on the maps had no impact on the answers to the two
questions.
R
R
D
A
L
D
A
L
By thinking of each bank and island as a vertex and each
bridge as an edge joining them Euler was able to model
the situation using the graph on the right. Hence, the
Königsberg puzzle is the same as asking if the graph has
an Euler path or Euler circuit.
Slay-age
Example:
• The Scooby Gang needs to patrol the following section of
town starting at Sunnydale High (labeled G). Draw a graph
that models this situation, assuming that each side of the
street must be checked except for those along the park. (Map
is from p. 206)
Example 2: (Exercise 21, pg 207) The
map to the right of downtown Kingsburg,
shows the Kings River running through
the downtown area and the three islands
(A, B, and C) connected to each other and
both banks by seven bridges. The
Chamber of Commerce wants to design a
walking tour that crosses all the bridges.
Draw a graph that models the layout of
Kingsburg.
Example 3:
The Kevin Bacon Game
(http://www.cs.virginia.edu/oracle/)
Euler’s Theorems
• Euler’s Theorem 1
(a) If a graph has any odd vertices, then
it cannot have an Euler circuit.
(b) If a graph is connected and every
vertex is even, then it has at least one
Euler circuit.
• Euler’s Theorem 2
(a) If a graph has more than two odd
vertices, then it cannot have an Euler
path.
(b) If a connected graph has exactly two
odd vertices then it has at least one
Euler path starting at one odd vertex
and ending at another odd vertex.
Example 4: Königsburg’s Bridges III (The Search For More Money)
Let us consider again the Königsburg Brdige puzzle as represented by the graph
below:
R
D
A
L
We have already seen that the puzzle boils down to whether this graph has
an Euler path and/or an Euler circuit. Does this graph have either?
Example 5: (Exercise 60, pg 214) Refer to Example 2. Is it possible to take a
walk such that you cross each bridge exactly once? Explain why or why not.
N
A
B
S
C
Example 6: Unicursal Tracings
Recall the routing problems presented on Tuesday:
•“Do these drawings have unicursal tracings? If so, are they open or closed?”
How might we answer these queries? Well, if we add vertices to the corners of the
tracings we can reduce the questions to asking whether the following graphs have Euler
paths (open tracing) and/or Euler circuits (closed tracing).
(a)
(b)
(c)
• Euler’s Theorem 3
(a) The sum of the degrees of all the
vertices of a graph equals twice the
number of edges.
(b) A graph always has an even number
of odd vertices.
A quick summary . . .
Number of odd vertices
Conclusion
0
Graph has Euler
circuit(s)
2
Graph has Euler
path(s) but no Euler
circuit
Graph has no Euler
path and no Euler
circuit
Impossible!
4, 6, 8, . . .
1, 3, 5, . . .
§5.6 Fleury’s Algorithm
• Euler’s Theorems give us a simple way
to see whether an Euler circuit or an
Euler path exists in a given graph, but
how do we find the actual circuit or
path?
• We could use a “guess-and-check”
method, but for a large graph this could
lead to many wasted hours--and not
wasted in a particularly fun way!
Algorithms
An algorithm is a set of procedures/rules that, when
followed, will always lead to a solution* to a
given problem.
• Some algorithms are formula driven--they arrive
at answers by taking data and ‘plugging-in’ to
some equation or function.
• Other algorithms are directive driven--they arrive
at answers by following a given set of directions.
Fleury’s Algorithm
• The Idea:
“Don’t burn your bridges behind you.”
(“bridges”: graph-theory bridges, not real world)
• When trying to find an Euler path or an Euler
circuit, bridges are the last edges we should
travel.
• Subtle point: Once we have traversed an
edge we no longer care about it--so by
“bridges” we mean the bridges of the part of
the graph that we haven’t traveled yet.
Example 1: Does this graph have an Euler circuit? If so, find
one.
A
B
D
C
E
F
Fleury’s Algorithm
1) Ensure the graph is connected and all the
vertices are even*.
2) Pick any vertex as the starting point.
3) When you have a choice, always travel
along an edge that is not a bridge of the yetto-be-traveled part of the graph.
4) Label the edges in the order which you
travel.
5) When you can’t travel anymore, stop.
* - This works when we have an Euler circuit. If we
only have a path, we must start at one of (two) the
odd vertices.
Example 2: Do the following drawings have unicursal tracings? If so,
label the edges 1, 2, 3, . . . In the order in which they can be traced.
Example 3: (Exercise 60, pg 214) The
map to the right of downtown Kingsburg,
shows the Kings River running through
the downtown area and the three islands
(A, B, and C) connected to each other and
both banks by seven bridges. The
Chamber of Commerce wants to design a
walking tour that crosses all the bridges.
Draw a graph that models the layout of
Kingsburg.
It was shown yesterday that it was
possible to take a walk in such that you
cross each bridge exactly once. Show
how.
N
A
B
S
C
Slay-age
Example:
• The Scooby Gang needs to patrol the following section of
town starting at Sunnydale High (labeled G). Suppose that
they must check each side of the street except for those
along the park. Find an optimal route for our intrepid demon
hunters to take.
Quiz 1, problem 2
North Bank (N)
B
A
C
South Bank (S)
Mathematics and the Arts?
• One of Euler’s 800+
publications
included a treatise
on music theory.
• Book was too mathy for most
composers--too
music-y for most
mathematicians
Mathematics and the Arts?
• While Euler’s
theories did not
catch on, a
relationship between
mathematics and
music composition
does exist in what is
called the golden
ratio.
Fibonacci Numbers
• The Fibonacci Numbers are
those that comprise the
sequence:
1, 1, 2, 3, 5, 8, 13, 21, . . .
• The sequence can be
defined by:
F1=1, F2=1;
Fn=Fn-1+Fn-2
• These numbers can be used
to draw a series of ‘golden’
rectangles like those to the
right.
Fibonacci Numbers
• The sequence of
Fibonacci Ratios fractions like 3/5, 5/8,
8/13 approach a
number called the
Golden Ratio
(≈0.61803398…)
The Golden Ratio
• Several of Mozart’s piano
sonatas make use of this
ratio.
• At the time such pieces
regularly employed a division
into two parts
1. Exposition and
Development
2. Recapitulation
• In Piano Sonata No. 1 the
change between parts
occurs at measure 38 of 100.
(which means that part 2 is
62 ≈ 0.618 x 100)
The Golden Ratio
• Another example in music is
in the ‘Hallelujah’ chorus in
Handel’s Messiah.
• The piece is 94 measures
long.
• Important events in piece:
1. Entrance of trumpets “King of Kings” occurs in
measures 57-58 ≈ (8/13) x
94
2. “The kingdom of glory…”
occurs in meas.
34-35 ≈ (8/13) x 57
etc, etc. . .
The Golden Ratio in Art
H
Approx. = 0.618 x H
The Golden Ratio in Art
H
Approx. = 0.618 x H
The Golden Ratio in Art
The Golden Ratio in Art
.618 x Ht.
0.618 x Width
The Golden Ratio in Art
.618 x Ht.
0.618 x Width