§5.2 5.3 Graphs and Graph Terminology

Graph Terminology•A path that starts and ends at the same vertex is called a circuit.. Example 6: The graph on the left has no Euler paths, but the one on the right has several... B

§5.2 - 5.3 Graphs and

Graph Terminology

“Liesez Euler, Liesez

maître à tous.”

- Pierre Laplace

Graphs consist of

 points called

vertices

 lines called edges

1 Edges connect two

is a graph List its vertices and

Example 2:

Flexo Bender Leela

Fry Amy

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).

QuickTime™ and a

TIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture.

Farnsworth Zoidberg

Flexo Bender

Amy Farnsworth

Now check out the graph below.

What can we say about it in comparison to

the previous figure?

QuickTime™ and a

TIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture.

Zoidberg

• 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.)

Moral of the Story

Graph Terminology

Graph Terminology

(ie - all the math-y jargon one could ask for)

 Adjacent Vertices are

two vertices that are

joined by an edge.

 Adjacent Edges are

two edges that

intersect at a vertex.

 The degree of a

vertex is the number of

Graph Terminology

A loop counts twice

toward the degree.

An odd vertex is a vertex of odd degree.

An even vertex is a vertex of even degree.

Graph Terminology

 A path is a sequence

of vertices such that

each vertex is adjacent

to the next In a

path, each edge can be

traveled only once

•The length of a path

is the number of edges

in that path.

Graph Terminology

•A path that starts

and ends at the same

vertex is called a

circuit .

•A graph is connected

if any two vertices

can be joined by a

path If this is not possible then the

disconnected

disconnected.

B H

S J

W K

Example 4:

1) Find a path from B to

K passing through W but not S.

2) Find a path from H to

Graph Terminology

path that travels

through every edge of the graph (once and

circuit that travels

through every edge of

a graph.

Example 6: The graph on the

left has no Euler paths, but

the one on the right has

several.

§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

L

D

What Euler realized was that most of the information

on the maps had no impact on the answers to the two questions

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

K ö nigsberg puzzle is the same as asking if the graph has

Example: Slay-

Slay-age

• 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 IIIKönigsburg’s Bridges III (The Search For More Money The Search For More Money)

Let us consider again the Königsburg Brdige puzzle as represented by the graph below:

R

A

L

D

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

S

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)

• 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

circuit(s)

path(s) but no Euler circuit

4, 6, 8, Graph has no Euler

path and no Euler circuit

1, 3, 5, Impossible!

§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!

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,

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

yet-to-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

Example: Slay-

Slay-age

• 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.

Mathematics and the

math-y for most

composers too

music-y for most

mathematicians

Mathematics and the

Fibonacci Numbers

• The Fibonacci Numbers

are those that comprise

• These numbers can be

used to draw a series

of ‘golden’ rectangles

like those to the

The Golden Ratio

• 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

The Golden Ratio in

Art

H

The Golden Ratio in

Art

H

The Golden Ratio in

Art

The Golden Ratio in

The Golden Ratio in