The Age of Euler
Chapter 10
Part 1
Lewinter & Widulski
The Saga of Mathematics
1
Leonhard Euler [1707-1783]



Euler is considered the
most prolific
mathematician in
history.
His contemporaries
called him “analysis
incarnate.”
“He calculated without
effort, just as men
breathe or as eagles
sustain themselves in
the air.”
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Leonhard Euler [1707-1783]
Euler was born in Basel, Switzerland, on
April 15, 1707.
 He received his first schooling from his
father Paul, a Calvinist minister, who had
studied mathematics under Jacob
Bernoulli.
 Euler's father wanted his son to follow in
his footsteps and, in 1720 at the age of
14, sent him to the University of Basel to
prepare for the ministry.

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Leonhard Euler [1707-1783]
At the age of 15, he received his
Bachelor’s degree.
 In 1723 at the age of 16, Euler completed
his Master's degree in philosophy having
compared and contrasted the philosophical
ideas of Descartes and Newton.
 His father demanded he study theology
and he did, but eventually through the
persuading of Johann Bernoulli, Jacob’s
brother, Euler switched to mathematics.

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Leonhard Euler [1707-1783]
Euler completed his studies at the
University of Basel in 1726.
 He had studied many mathematical works
including those by Varignon, Descartes,
Newton, Galileo, von Schooten, Jacob
Bernoulli, Hermann, Taylor and Wallis.
 By 1727, he had already published a
couple of articles on isochronous curves
and submitted an entry for the 1727
Grand Prize of the French Academy on the
optimum placement of masts on a ship.

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Leonhard Euler [1707-1783]
Euler did not win but instead received an
honorable mention.
 He eventually would recoup from this loss
by winning the prize 12 times.
 What is interesting is that Euler had never
been on a ship having come from
landlocked Switzerland.
 The strength of his work was in the
analysis.

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The 18th Century
The rise of scientific and mathematical
journals of the preceding century was the
quickest way of making new discoveries
known.
 This outgrowth of the printing revolution
of the 15th century accelerated the pace of
mathematical and scientific progress by
transmitting new ideas in a timely manner.


Similar to the growth of the information age.
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The 18th Century
The 18th century was still an age when no
man could consider himself educated
without a knowledge of mathematics,
for on mathematics all knowledge was
based.
 The universities were not the principal
centers of research.
 This nurturing was done by the various
royal academies supported by generous
rulers, like, Fredrick the Great of Prussia
and Catherine the Great of Russia.

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The 18th Century
These academies gave Euler the chance to
be the most prolific mathematician of all
time.
 They were research organizations which
paid their leading members to produce
scientific research.
 Of course, the academicians were paid to
produce results but once the rulers got a
reasonable return on their investment,
Euler, Lagrange, and the others were free
to do as they pleased.

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The 18th Century
The rulers of the 18th century let science
take its course.
 The first practical problem of this age was
the control of the seas.
 This meant accurate navigation techniques
which ultimately requires determining
one’s position while out at sea.
 This position is determined by observing
the heavens.

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The 18th Century
After Newton’s universal law suggested
that the position of the planets and the
phases of the Moon could be calculated for
centuries in advance, those wanting to
rule the seas started number crunching.
 The Moon offers a particularly difficult
problem involving three bodies
attracting one another; the Moon, the
Earth and the Sun.


Euler was the first to derive an approximate
solution.
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The Saga of Mathematics
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Leonhard Euler [1707-1783]
Euler eventually obtained royal
appointments in several European courts
including Russia and Germany (under
Frederick the Great).
 Two of Euler’s friends, Daniel and Nicholas
Bernoulli, encouraged Catherine I (wife of
Peter the Great) to appoint him a position
in the medical section at St. Petersburg.
 Euler quickly attended lectures on
medicine at Basel in hopes of obtaining
the post, which he received in 1727.

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Leonhard Euler [1707-1783]
Even in physiology, Euler could not keep
away from mathematics.
 The physiology of the ear suggested an
investigation of sound, which in turn led to
the propagation of waves.
 Euler eventually wrote an article on
acoustics, which went on to become a
classic.
 Quantity as well as quality
characterized Euler’s work.

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Leonhard Euler [1707-1783]
Upon Nicholas Bernoulli’s death, Euler was
appointed as head of the Natural
Philosophy department.
 In 1733, Daniel Bernoulli returned to
Switzerland and Euler, at the age of 26,
was appointed to senior chair of
mathematics.
 The publication of many articles and his
book Mechanica (1736-37) – a two
volume book on mechanics – started him
on the way to major mathematical work.

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Euler’s Mechanica (1736)
First textbook in which Newton’s dynamics
of the mass point was developed with
analytical methods.
 Followed by the Theoria motus corporum
solidorum seu rigidorum (1765) in which
the mechanics of solid bodies was similarly
treated.
 The later contains the “Eulerian” equations
for a body rotating about a point.

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A Famous Tale
Euler and the Atheist
Catherine the Great had Denis Diderot, a
French philosopher and editor of the great
French Encyclopédie, visit her Court.
 Diderot an atheist tried to convert the
courtiers to atheism.
 Fed up with Diderot, Catherine asked Euler
to puzzle him.
 Diderot was informed that a learned
mathematician was in possession of an
algebraic proof of the existence of God.

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Euler and the Atheist
Diderot consented to hear it even though
he knew nothing about mathematics.
 As the story goes, Euler approached
Diderot and said, “Monsieur,

ab
x
n
n
donc Dieu existe; répondez!”

That is, “Sir,
Lewinter & Widulski
a  bn
 x,
n
hence God exists; reply!”
The Saga of Mathematics
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Euler and the Atheist
This sounded like sense to Diderot.
 He was humiliated by the uncontrolled
laughter.
 Diderot asked permission to return to
France at once, which was granted.
 Of course, Euler’s argument was nonsense
but Diderot didn’t see it.
 Euler would eventually meet his match in
arguments with Voltaire.

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Leonhard Euler [1707-1783]
Euler had a phenomenal memory.
 As a boy, Euler memorized Virgil’s Aeneid
and could recite it flawlessly the rest of his
life.
 Euler not only memorized the first 100
prime numbers but also their squares,
cubes, fourth, fifth and sixth powers!
 He could also perform difficult calculations
mentally, some of which required him to
retain in his head 50 places of accuracy.

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Leonhard Euler [1707-1783]
Euler’s constant outflow of ideas is
legendary.
 It is said that he would write a
mathematical paper in the half hour
between the first and second calls for
dinner.
 He published three monumental works on
analysis, and also wrote on algebra,
arithmetic, mechanics, music, chemistry,
and astronomy.

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Leonhard Euler [1707-1783]
In 1741, Euler was invited by Frederick
the Great of Prussia to come to Berlin to
teach and do research.
 In Berlin, Euler published his Introductio in
Analysin infinitorum (1748).
 This was followed by Institutiones calculi
differentialis (1755) and the three volume
Institutiones calculi integralis (1768-74).


Instantly became classics.
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Euler’s Analysis Infinitorum

Divided into two
parts:



Algebra, theory of
equations and
trigonometry
Analytical geometry
It contains the
expansion of various
functions in series and
the summation of
certain series.
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Euler’s Analysis Infinitorum
ei + 1 = 0
He pointed out that an infinite series
cannot be safely added unless it is
convergent.
 Although he recognized this necessity for
dealing with series, he often failed to
observe it in much of his own work.
 He introduced the current abbreviations
for the trigonometric functions, and
showed that ei = cos  + i sin .

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Euler’s Analysis Infinitorum
Euler showed that the general equation of
second degree
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
represents the various conic sections.
 He extended the application of analytical
geometry to three dimensions, where he
found general forms for the equations of
different solids.


A circle centered at the origin is given by the
equation x2 + y2 = r2 and a sphere centered at
the origin is given by x2 + y2 + z2 = r2.
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Euler’s Institutiones calculi integralis
A thorough investigation of integrals.
 It includes Taylor’s theorem with many
applications.
 The Beta and Gamma functions were
invented by Euler and he uses them as
examples of integration.
 As well as investigating double integrals,
Euler considered ordinary and partial
differential equations in this work.

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Leonhard Euler [1707-1783]
Although he lost the sight in one eye in
1735 and the other eye in 1766, nothing
could interrupt his enormous productivity.
 In 1770 Euler published his Vollständige
Anleitung zur Algebra.



A French translation with numerous and
valuable additions by Lagrange appeared in
1774.
In this text, Euler proves xn + yn = zn is
impossible for integers x, y, z, n=3 and n=4.
(Fermat’s Last Theorem)
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Leonhard Euler [1707-1783]
In 1744 appeared Euler’s Methodus
inveniendi lineas curvas maximi minimive
proprietate gaudentes.
 He includes solutions to the classic
problems on isoperimetrical curves, the
brachistochrone in a resisting medium,
and the theory of geodesics.
 It was this that lead him to the calculus
of variations, a sort of generalization of
calculus.

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Other works by Euler

His most important works on astronomy in
which he attacked the problem of three
bodies are:




Theoria Motuum Planetarum et Cometarum
(1744).
Theoria Motus Lunaris (1753)
Theoria Motuum Lunae (1772)
His three volume work on optics Dioptrica
(1769-71).
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Other works by Euler
In 1739 appeared his new theory of music
Tentamen novae theoriae musicae which,
it is said, was too musical for
mathematicians and too mathematical
for musicians.
 Lettres a une princess d'Allemagne sur
divers sujets de physique & de philosophie
(1760-61) were composed to give lessons
in physics, mechanics, optics, astronomy
and sound.

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Euler’s Letters to a German Princess
During Euler’s stay in Berlin (1741-66), he
was asked to provide some tutoring in
Natural Philosophy (elementary science)
to Princess d'Anhalt Dessau, a niece of
Frederick the Great.
 These lectures were published in several
volumes entitled Letters to a German
Princess (1760-61), and for half a century
they remained a standard treatise on the
subject.

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Euler’s Letters to a German Princess
They became immensely popular and were
circulated in seven languages.
 William Dunham says the they are one of
history’s finest example of “popular
science.”
 What we call Venn diagrams first
appears in Euler’s Letters.
 Venn himself first called them "Eulerian
Circles", but then somehow managed to
get them called Venn Diagrams.

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Leonhard Euler [1707-1783]
Many other results of Euler can be found
in his smaller papers.
 Some of the better known results are:







Euler’s Polyhedron Formula: V – E + F = 2.
The Euler Line of a Triangle.
Euler’s constant  = 0.577215664901532….
Euler's theorem (also known as the FermatEuler theorem).
Euler’s pentagonal formula for partitions.
Eulerian graphs
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Leonhard Euler [1707-1783]
Euler was in a sense the creator of
modern mathematical expression.
 In terms of mathematical notation, Euler
was the person who gave us:


 for pi
i for 1
 y for the change in y
 f(x) for a function


 for summation
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Leonhard Euler [1707-1783]
To get an idea of the magnitude of Euler’s
work it is worth noting that:
 Euler wrote more than 500 books and
papers during his lifetime – about 800
pages per year.
 After Euler’s death, it took over forty
years for the backlog of his work to
appear in print.


Approximately 400 more publications.
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Leonhard Euler [1707-1783]
He published so many mathematics
articles that his collected works Opera
Omnia already fill 73 large volumes –
tens of thousands of pages – with more
volumes still to come.
 More than half of the volumes of Opera
Omnia deal with applications of
mathematics – acoustics, engineering,
mechanics, astronomy, and optical devices
(telescopes and microscopes).

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Leonhard Euler [1707-1783]
His publications account for one-third of
all the technical articles published in 18th
century Europe.
 He lost his sight sometime after 1766, yet
he continued his research at his usual
energetic pace while his students wrote it
down.
 So, what areas of math did he enrich and
expand?

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Leonhard Euler [1707-1783]
The question is what field of math did he
not enrich and expand!
 Not only did he contribute substantially to







Calculus
Geometry
Algebra
Mechanics
and Number Theory
He invented several fields.
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Leonhard Euler [1707-1783]
Euler was the father of thirteen children
(all but five died very young) and still
found time to become the father of an
important branch of mathematics, known
today as graph theory.
 Important in such fields as computer
science, networking, operations research,
physics and chemistry.
 Euler became the father of graph theory
after solving the “Seven Bridges of
Königsberg” problem.

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The Bridges of Königsberg Problem
In 1736, Euler published his solution to
the problem known as the Seven Bridges
of Königsberg in a paper Solutio
problematis ad geometriam situs
pertinentis.
 This paper is considered to be the earliest
application of graph theory or topology.
 It is also regarded as one of the first
topological results in geometry; that is, it
does not depend on any measurements.

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The Seven Bridges of Königsberg
A
D
B
C
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The Bridges of Königsberg Problem
The Problem: Find a route that crosses
each bridge exactly once and returns to
where it starts.
 Euler observed that it could not be done!
 Each landmass has an odd number of
bridges.
 A traveler departing, returning, departing,
etc. an odd number of times would wind
up departing on the last bridge, making it
impossible to return to the point of origin.

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The Bridges of Königsberg Problem
Let’s consider this gem of thinking one
more time.
 Number the bridges contiguous with
landmass A, 1, 2, and 3.
 If one starts the trip by departing A on
bridge #1, they must return on bridge #2
or #3, leaving only one more bridge.
 They must depart on the bridge not yet
traveled on – and that makes all the
difference!
 You cannot end your trip on landmass A.

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The Bridges of Königsberg Problem



Observe that the sizes
of the land masses as
well as the lengths
and shapes of the
bridges are irrelevant.
Thus, you can redraw
the diagram above
with the landmasses
as dots and the
bridges as lines.
See the Figure.
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Leonhard Euler [1707-1783]
Notice the irrelevance of the weird shapes
of the bridges meeting at B.
 The lengths of the lines and the precise
locations of the dots are also unimportant.
 Euler considered this problem in the
context of Leibniz’s desire for a type of
geometry that doesn’t involve the concept
of a metric such as length or distance.


This is topology or rubber-sheet geometry –
The problem is the same if you draw it on
rubber and stretch it.
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Euler’s letter to Giovanni Marinoni

“This question is so banal, but seemed to me
worthy of attention in that neither geometry, nor
algebra, nor even the art of counting was
sufficient to solve it. In view of this, it occurred to
me to wonder whether it belonged to the
geometry of position, which Leibniz had once so
much longed for. And so, after some deliberation,
I obtained a simple, yet completely established,
rule with whose help one can immediately decide
for all examples of this kind whether such a
round trip is possible.”
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§1: Graphs in Graph Theory
Today the problem is solved by looking at
a graph, or a network, with points
representing the land masses and lines
representing the bridges.
 We define a graph as follows:
 A graph G is a collection of dots (called
vertices), and a collection of lines (called
edges), each line rendering a pair of
vertices adjacent.


That is, the edge links the two vertices.
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Definition of a Graph
A graph G=(V,E)
consists of:
 a set V = V(G) of
vertices or
nodes, and
 a set E = E(G) of
edges: unordered
pairs of distinct
elements u,v  V.

Lewinter & Widulski
Visual Representation
of a Simple Graph
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Example of a Graph

Let V be the set of states in the north
eastern part of the U.S.:


V={ME, NH, VT, MA, RI, CT, NY, NJ, PA}
Let E={{u,v}|u adjoins v}
={{ME,NH},{NH,VT},{NH,MA},
{VT,MA},{VT,NY},{NY,MA},
{NY,CT},{NY,NJ},{NY,PA},
{MA,RI},{MA,CT},{CT,RI},
{NJ,PA}}
VT
NY
PA
NH
ME
MA
CT
RI
NJ
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Example of a Graph (continued)

The specific layout, or representation, of
the graph doesn’t matter, as long as the
adjacencies and non-adjacencies are
preserved.


CT is not that close to NJ!
Note: There is an edge
between two vertices if
the share a border.
VT
NY
PA
NH
ME
MA
CT
RI
NJ
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Directed Graphs
A directed graph or digraph D = (V,A)
consists of a set V of nodes together with
a set A of ordered pairs of distinct nodes
in V called directed edges or arcs.
 E.g.: V = species in an ecosystem,
A={(x,y) | x preys on y}

A food web
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Variations
There are several variations of graphs
which deserve mention.
 Note that the definition of a graph permits
no loop, i.e., no edge joining a point to
itself.
 In a multigraph, no loops are allowed but
more than one edge can join two nodes;
these are called multiple edges.
 If both loops and multiple edges are
permitted, we have a pseudograph.

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Parallel
edges
Multigraphs
We will not consider graphs in which a
single pair of vertices are linked by more
than one edge, as in the graph of the
Königsberg Bridge Problem, where
vertices A and B are linked by two edges.
 Such graphs are called multigraphs and
are important in certain transportation
problems.


For example, vertices or nodes are cities and
the edges are segments of major highways.
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Directed Multigraphs
Like directed graphs, but there may be
more than one arc from a node to another.
 A directed multigraph G=(V, E, f ) consists
of a set V of vertices, a set E of edges,
and a function f:EVV.
 E.g., V=web pages,
E=hyperlinks. The WWW is
a directed multigraph...

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Pseudographs
Like a multigraph, but edges connecting a
node to itself are allowed.
 A pseudograph G=(V, E, f ) where
f:E{{u,v}|u,vV}. Edge eE is a loop
if f(e)={u,u}={u}.
 E.g., nodes are campsites
in a state park, edges are
hiking trails through the woods.

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Types of Graphs: Summary

Keep in mind this terminology is not fully
standardized...
Term
Edge Type
Multiple
Edges ok?
Self-loops
ok?
Graph
Undir.
No
No
Multigraph
Undir.
Yes
No
Pseudograph
Undir
Yes
Yes
Digraph
Directed
No
Yes
Directed
Multigraph
Directed
Yes
Yes
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Adjacency
Let G be a graph with edge set E.
 Let eE be the edge joining u and v, that
is, e = {u,v} or simply e = uv.
 We say:
 u, v are adjacent / neighbors /
connected.
 Edge e is incident with vertices u and v.
 Edge e connects or joins u and v.

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Degree of a Vertex
Let G be a graph and vV a vertex.
 The degree of vertex v, denoted deg(v),
is the number of edges incident with v.
(Except that any self-loops are counted
twice.)
 A vertex with degree 0 is isolated.
 A vertex of degree 1 is an endpoint,
endnode, or endvertex.

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Degree Sequence
If G is a graph with n nodes, the degree
sequence (d1, d2, d3, …, dn) of G is the
non-increasing sequence of degrees of the
nodes of G.
 For example, (2,2,2,1,1) is the degree
sequence for P5 or the graph G below.

G
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P5
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§2: Graph Theory Concepts

The graph G below will be used to
demonstrate several concepts in graph
theory.
a
G
b
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d
c
g
e
f
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h
j
i
59
Degree of a Vertex

The degree of a vertex is the number of
edges touching it (technically, incident
with it). a
d
c
G
b

g
e
f
h
j
i
Thus, the degree of vertex g in graph G
above is 4.

This is written as deg(g)=4.
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Notation
Graphs are usually identified by capital
letters and the vertices by lowercase
letters.
 Edges may also be labeled using small
letters, but the common practice is to
label an edge using the letters of the two
vertices it is incident with.
 The rightmost edge in graph G, for
example, may be referred to as edge hj.

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Vertex Set and Edge Set
The set of vertices and the set of edges of
a graph G are denoted V(G) and E(G),
respectively.
 We will use the convention that n and e
represent the cardinalities (i.e., sizes) of
the vertex set and edge set, respectively.
 For the above graph,


V(G) = {a, b, c, d, e, f, g, h, i, j}
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Vertex Set and Edge Set
In this case, graph G has ten vertices, so
n=10.
 Also


E(G) = {ac, be, cd, cg, dh, ef, eg, fg, gh, hi,
hj}
G has eleven edges, therefore, e = 11.
 Vertices a, b, i and j have degree 1, and
are therefore called endvertices.

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Handshaking Theorem
Euler established the following interesting
fact, important enough to be called a
theorem.
 Theorem: The sum of the degrees of the
vertices of a graph equals twice the
number of edges.
 In other words, let G be a graph with
vertex set V and edge set E. Then


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 deg( v)  2 E
vV
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Handshaking Theorem
The proof is easy! Each edge contributes
one to each of the degrees of the two
vertices to which it is adjacent.
 Therefore the degree sum is twice the
number of edges.
 As a consequence, the sum of the degrees
of any graph must be an even number.
 Corollary: A graph has an even number
of vertices of odd degree.

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Directed Adjacency
Let G be a digraph, and let e be an edge
of G from u to v, that is e = {u,v} = uv.
 Then we say:






u is adjacent to v, v is adjacent from u
e comes from u, e goes to v.
e connects u to v, e goes from u to v
the initial vertex of e is u
the terminal vertex of e is v
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Directed Degree

Let G be a digraph, and v a vertex of G.



The in-degree of v, deg(v), is the number of
edges going to v.
The out-degree of v, deg(v), is the number
of edges coming from v.
The degree of v, deg(v)=deg(v)+deg(v), is
the sum of v’s in-degree and out-degree.
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Directed Handshaking Theorem
Let G be a digraph with vertex set V and
edge set E.
 Then:

1
deg (v)   deg (v)   deg( v)  E

2 vV
vV
vV



Note that the degree of a node is
unchanged by whether we consider its
edges to be directed or undirected.
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§3: Special Classes of Graphs
Complete graphs Kn
 Cycles Cn
 Regular Graphs
 Paths Pn
 Wheels Wn
 Hypercubes or n-Cubes Qn
 Bipartite graphs
 Complete bipartite graphs Km,n
 The n-dimensional Octahedron

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Complete Graphs

For any positive integer n, a complete
graph on n vertices, Kn, is a graph with n
nodes in which every node is adjacent to
every other node.
K1
K2
K4
K3
Note: Kn has
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n 1
i 
i 1
K5
n(n  1)
2
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K6
edges.
70
Cycles

For any n3, a cycle on n vertices, Cn, is a
graph where V={v1,v2,… ,vn} and
E={{v1,v2},{v2,v3},…,{vn1,vn},{vn,v1}}.
C3
C4
C5
C6
C7
C8
How many edges are there in Cn?
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Regular Graphs
A graph in which each vertex has the
same degree is called regular.
 If the common degree is r, we call the
graph r-regular.
 Note that each vertex of a cycle has
degree two. Thus, the cycles Cn are 2regular.
 The complete graphs Kn are (n–1)-regular.


Can you draw a 3-regular graph on six nodes?
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Paths

Another very important class of graphs
are paths, denoted Pn, where n is, once
again, the number of vertices in the path.
P5.
P1
P2
P3
P4
P5
P6
How many edges are there in Pn?
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Wheels

For any n3, a wheel Wn, is a graph
obtained by taking the cycle Cn-1 and
adding one extra vertex vhub and n-1 extra
edges {{vhub,v1}, {vhub,v2},…,{vhub,vn-1}}.
W4
W5
W6
W7
W8
W9
How many edges are there in Wn?
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Hypercubes (n-cubes)

For any positive integer n, the hypercube
Qn is a simple graph consisting of two
copies of Qn-1 connected together at
corresponding nodes. Q0 has 1 node.
Q0
Q1
Q2
Q3
Q4
Number of vertices: 2n.
Number of edges: Exercise to try!
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Bipartite Graphs

A bipartite graph G is a graph whose
vertex set can be partitioned into two
subsets V1 and V2 such that every edge of
G joins V1 with V2.
Q3
Q3
Q3
Q3
Theorem: A graph is bipartite iff all its cycles are even.
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Complete Bipartite Graphs

A complete bipartite graph, Km,n, is a
bipartite graph which contains every edge
joining V1 and V2.
K2,3
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K4,4
77
The n-dimensional Octahedron
Draw a regular polygon with 2n sides.
 Join two nodes by an edge if they are not
directly opposite each other.

The 3-dimensional
Octahedron
The 4-dimensional Octahedron
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§4: Graph Operations
Subgraphs
 Unions
 Complement
 Join (omitted)
 Product (omitted)
 Composition (omitted)

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Subgraphs

A subgraph of a graph G=(V,E) is a
graph H=(W,F) where WV and FE.
G
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H
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Subgraph Example

The hypercube Q3 is a subgraph of the
complete bipartite K4,4.
Q3
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Graph Unions

The union G1G2 of two simple graphs
G1=(V1, E1) and G2=(V2,E2) is the simple
graph (V1V2, E1E2).
G1
G2
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G1G2
82
Graph Complement

The complement G of a graph G has
V(G) has its vertex set, but two vertices
are adjacent in G if and only if they are
not adjacent in G.
G
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G
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§5: Graph Representations &
Isomorphism

Graph Representations:




Adjacency Lists
Adjacency Matrices
Incidence Matrices
Graph Isomorphism:

Two graphs are isomorphic if and only if they
are identical except for their node names.
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Adjacency Lists

A table with 1 row per
vertex, listing its
adjacent vertices.
a
f
b
e
c
d
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Vertex Adjacent Vertices
a
b, f
b
c
d
e
a, d, f
d
b, c, f,
f
a, b, d
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85
Directed Adjacency Lists

1 row per node, listing
the terminal nodes of
each edge incident
from that node.
a
f
b
e
c
Vertex Adjacent Vertices
a
b, f
b
c
d
e
d
f
b, d
c
d
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Adjacency Matrix

Matrix A=[aij], where
aij is 1 if {vi, vj} is an
edge of G, 0
otherwise.
a
b
c
d
e
f
a
0
1
0
0
0
1
b
1
0
0
1
0
1
c
0
0
0
1
0
0
d
0
1
1
0
0
1
e
0
0
0
0
0
0
f
1
1
0
1
0
0
a
f
b
e
c
d
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Adjacency Matrix
Notice that the sum of a row (or column)
is equal to the degree of that vertex.
 Hence the isolated vertex e appears as a
row and column of all zeros.
 For a simple graph with no self-loops, the
adjacency matrix must have 0s on the
diagonal.
 For an undirected graph, the adjacency
matrix is symmetric.

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Incidence Matrix

The incidence matrix
of a graph has a row
for each vertex and
column for each edge,
and (v, e)=1 if vertex
v and edge e are
incident, 0 otherwise.


a
First defined by the
physicist Kirchhoff
(1847).
Each column contains
exactly two ones.
Why?
Lewinter & Widulski
4
1
5
d
3
b
2
c
a
b
c
d
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1
2
3
4
5
1
1
0
0
0
1
1
0
0
0
1
1
1
0
0
1
0
1
0
1
89
Graph Isomorphism

Formal definition:



Simple graphs G1=(V1, E1) and G2=(V2, E2) are
isomorphic if and only if there exists a
bijection f:V1V2 such that for all a,b  V1, a
and b are adjacent in G1 if and only if f(a) and
f(b) are adjacent in G2.
f is the “renaming” function that makes the
two graphs identical.
Definition can easily be extended to other
types of graphs.
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Graph Invariants under Isomorphism

Necessary but not sufficient conditions for
G1=(V1,E1) to be isomorphic to
G2=(V2,E2):



|V1|=|V2|, |E1|=|E2|.
The number of vertices with degree n is the
same in both graphs.
For every proper subgraph g of one graph,
there is a proper subgraph of the other graph
that is isomorphic to g.
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Isomorphism Example

If isomorphic, label the 2nd graph to show
the isomorphism, else identify difference.
d
b
b
a
d
a
c
e
c
e
f
f
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Are These Isomorphic?

If isomorphic, label the 2nd graph to show
the isomorphism, else identify difference.
* Same # of
nodes
* Same # of
edges
* Different
# of nodes of
degree 2!
(1 versus 3)
a
b
d
c
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Self Complementary Graphs

The self-complementary graph is
isomorphic with its complement.
~
=
G

G
Show that P4 is self-complementary.
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§6: Walks, Trials, and Paths

A walk of a graph G is an alternating
sequence of nodes and edges

v0, e1, v1, e2, v2, e3, v3, …, vn-1, en, vn
beginning and ending with nodes, such
that each edge is incident with the two
nodes immediately preceding and
following it.
 This walk, called a v0-vn walk, joins v0 and
vn and may also be denoted v0, v1, v2,
v3,…, vn-1, vn.

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Walks, Trials, and Paths
It is a closed walk if v0=vn, and is open
otherwise.
 It is a trial if all edges are distinct.
 It is a path if all the nodes (and
necessarily all the edges) are distinct.
 A closed path, n≥3, is a cycle.
 The length of a walk, trail or path is the
number of edges that occur in it.

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Walks, Trials, and Paths Examples

In G:



befeg is a walk which is not a trail.
cgfegh is a trail which is not a path.
acghi is a path and cdhgc is a cycle.
a
G
b
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c
g
e
f
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h
j
i
97
Connected Graphs


We will study graphs that are connected, that is,
there is a way to travel between any two vertices
by traversing a sequence of consecutive edges
between them.
For example, in the graph G below, you can
travel from vertex b to vertex d by traversing the
consecutive edge sequence be, eg, gc, cd.
a
d
c
G
b
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g
e
f
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j
i
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Connectedness
In other words, there is a path in the
graph whose end points are b and d.
 This path is called a b-d path.
 The vertices of this path form a sequence
in which consecutive members are
adjacent.



Note: there is another b-d path with vertices
b, e, g, h and d.
This is useful if the graph is an airline
graph and the airport in city c is closed.
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Connectedness

The traveler can be rerouted from city b to
city d by flying from g to h instead of from
g to c.


The same logic would apply if c were a
telephone exchange that is malfunctioning.
The reason we have travel options is that
graph G contains cycles, namely C3, with
vertices e, f and g, and C4, with vertices c,
d, g and h.
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Paths in Directed Graphs
Same as in undirected graphs, but the
path must go in the direction of the
arrows.
 In the digraph to
a
the right abdc is a
f
b
path.

e
c
d
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Connected Graphs
A graph G is connected if every pair of
nodes are connected by a path.
 A maximal connected subgraph of G is
called a connected component or just a
component of G.
 A disconnected graph has at least two
components.

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Cutpoints and Bridges
A cutpoint , or cut node, of a graph G is
a node whose removal increases the
number of components of G.
 An edge of a graph G is a bridge if its
removal increases the number of
components of G.

v1
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v3
v4
103
Directed Connectedness
A digraph D is strongly connected if
there is a directed path from any node of
D to any other node of D.
 It is weakly connected if the underlying
undirected graph (i.e., with edge
directions removed) is connected.
 Note strongly implies weakly but not viceversa.

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Connectivity

The connectivity κ = κ(G) of a graph G is
the minimum number of nodes whose
removal results in a disconnected or trivial
graph.


The connectivity of a disconnected graph is 0,
while the connectivity of a graph with a
cutnode is 1.
The complete graph Kn cannot be disconnected
by removing any number of nodes, but the
trivial graph results after removing n – 1
nodes; thus, κ(Kn) = n – 1.
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Edge-Connectivity

The edge-connectivity κ' = κ'(G) of a
graph G is the minimum number of edges
whose removal results in a disconnected
or trivial graph.


Thus κ'(K1) = 0, and the edge-connectivity of a
disconnected graph is 0, while the connectivity
of a graph with a bridge is 1.
κ'(Kn) = n – 1.
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§7: Planar Graphs
A graph is planar if it can be drawn in the
plane in such a way that the edges do not
intersect.
 For example, the graph K4 is planar.

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Five Points in the Plane
Can five points in the plane be joined by
lines in such a way that the lines do not
cross?
 In other words, is the graph K5 planar?
 The answer is NO!

y
K5 minus
an edge is
planar.
x
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Water, Gas, and Electricity

Lines from the water, gas, and electric
utilities are to be connected to three
houses A, B, and C. Can this be done in
such a way that the lines do not cross?
Lewinter & Widulski
A
B
C
W
G
E
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Water, Gas, and Electricity
This is equivalent to asking if the graph
K3,3 is planar.
 The answer is NO!
 Again this is almost true, but not quite.
 If we remove a single edge from K3,3 it
becomes planar, but however we try to
draw the last edge it will cross another
edge.
 Therefore, both K5 and K3,3 are not planar.

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Euler Characteristic
If a finite graph G is planar, it will have V
nodes, E edges, and a certain number of
faces F (the faces are the regions enclosed
by the edges. If G is drawn in the plane,
the region outside G is counted as a face).
 Theorem: If a graph G is planar,
then V – E + F = 2.


The quantity V – E + F is called the Euler
characteristic of G.
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Euler’s Formula

For any convex polyhedron,
V–E+F=2




Examples






Lewinter & Widulski
V = Vertices
E = Edges
F = Faces
Tetrahedron: V=4, E=6, F=4
Cube: V=8, E=12, F=6
Octahedron: V=6, E=12, F=8
Dodecahedron: V=20, E=30,
F=12
Icosahedron: V=12, E=30, F=20
BuckyBall: V=60, E=90, F=32
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Proof of Euler’s Formula
Proof by induction
 If no edges, its an isolated vertex. So
V=1, E=0, F=1
 Else choose any edge



If it connects two vertices, contract it. This
reduces V by 1 and E by 1
Else the edge must separate two faces (Jordan
curve). Remove it. Reduces F by 1 and E by 1.
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Euler Formula Example 1

For the graph K4,




V=4
E=6
F=4
So V – E + F = 2.
1
2
3
4
“the outside”
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Euler’s Formula Example 2

Show V – E + F = 2 for the dodecahedron.
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Non-Planar Graphs
We can use the previous theorem to prove
that certain graphs are not planar.
 First notice that if every cycle of a finite
planar graph G contains at least k edges,
then since each edge occurs on exactly
two faces, we have the inequality kF ≤
2E.

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Example 1

The complete graph K5 is not planar.






Notice that for this graph, V = 5 and E = 10.
Each cycle of K5 contains at least 3 edges.
Since V – E + F = 2, implies F = 7 if K5 is
planar.
By the inequality kF ≤ 2E.
21 = 3F ≤ 2E = 20.
Contradiction!
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Example 2

The complete bipartite graph K3,3 is not
planar.






Notice that V = 6 and E = 9.
So using Euler’s formula V – E + F = 2, implies
F = 5 if K3,3 is planar.
Each cycle of K3,3 contains at least 4 edges.
By the inequality kF ≤ 2E.
20 = 4F ≤ 2E = 18.
Contradiction!
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K3,3
118
Kuratowski’s Theorem

If G is a finite graph, then the following
conditions are equivalent:



G is not planar.
G contains a homeomorph of K5 or K3,3.
A homeomorph means that the nodes of
the graph are identified with the nodes of
K5 or K3,3 and the edges are identified with
disjoint paths.
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Homeomorphic Graphs

Two graphs, G and H are defined to be
homeomorphic if you can make one
graph into the other by inserting nodes of
degree 2.

Two graphs are homeomorphic if they are
isomorphic “up to vertices of degree 2”.
A homeomorph of K4.
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§8: Traversability
Euler’s negative solution of the Königsberg
Bridge Problem constituted the first
publicized discovery of graph theory.
 The abstraction of the problem to that of
one using a graph becomes:
 Given a graph G, is it possible to find a
walk that traverses each edge exactly
once, goes through all nodes, and ends at
the starting point?

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Eulerian Graphs
A graph for which this is possible is called
Eulerian.
 An Eulerian graph contains an Eulerian
circuit which is a closed trail containing
all the nodes and edges.
 Theorem: The following statements are
equivalent for a connected graph G:




G is Eulerian.
Every node of G has even degree.
The set of edges of G can be partitioned into
cycles.
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Eulerian Graphs

Corollary: Let G be a connected graph
with exactly 2 nodes of odd degree. The G
has an open trail containing all nodes and
edges of G (which begins at one odd node
and ends at the other).
Can you draw the figure at the right
without lifting your pencil off the paper?
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Fleury’s Algorithm

This algorithm will find an Eulerian circuit
or trail on a finite graph G, if such a circuit
or trail exist. If the algorithm terminates
without producing an Eulerian circuit or
trail, then G does not have an Eulerian
circuit or trail.


Beginning with any edge, choose edges so as
to give a trail in G. Erase edges as they are
chosen, and also erase any isolated nodes
which may occur.
Never choose an edge which is a bridge unless
there is no alternative.
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The 3-dimensional Octahedron

The 3-dimensional Octahedron is Eulerian.
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Other Examples
The complete graph Kn is Eulerian if and
only if n is odd (because the degree of
each node of Kn is n – 1).
 The graph of the n-cube is Eulerian if and
only if n is even (because the degree of
each node of the graph of the n-cube is
n).
 The graph of the n-dimensional
octahedron is always Eulerian (because
the degree of each node of this graph is
2n – 2, which is always even).

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Sona Sand Drawings



Sona drawings are
networks that are
drawn in the sand
without lifting the
finger or retracing
any line segments.
Tradition among
the Chokwe in
southern-central
Africa.
WWW links
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Hamiltonian Graphs
Sir William Hamilton suggested a class of
graphs which bear his name when he
asked for the construction of a cycle
containing every vertex of a
dodecahedron.
 If a graph G has a spanning cycle Z, then
G is called a Hamiltonian graph and Z a
Hamiltonian cycle.

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Round-the-World Puzzle

Can we traverse all the vertices of a
dodecahedron, visiting each once?
Dodecahedron
Puzzle
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Equivalent
Graph
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129
The 3-dimensional Octahedron

The 3-dimensional Octahedron is
Hamiltonian.
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Other Examples



The complete graph Kn is always Hamiltonian
(because this graph may be drawn by drawing a
regular polygon with n sides, and connecting all
pairs of nodes).
The graph of the n-cube is always Hamiltonian (if
we label the vertices with binary vectors of length
n, the Standard Gray Code gives a Hamiltonian
cycle).
The graph of the n-dimensional octahedron is
always Hamiltonian (remember that we draw this
graph by drawing a regular polygon with 2n
sides, and connecting all pairs of nodes by an
edge except those which are directly opposite).
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The Two-Way Street Problem
Consider any connected array of streets.
 Construct an associated graph by letting
each street corner or intersection
correspond to a node and each street
correspond to an edge.
 Double each edge.

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The Two-Way Street Problem
This is clearly Eulerian, since each node has even degree.
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The Chinese Postman Problem
A postman must cover a certain route,
passing along all streets of the route at
least once and returning to his starting
point.
 He wishes to do this in such a way that
the total distance traveled is a minimum.



If the graph corresponding to the arrays of
streets is Eulerian, then any Eulerian circuit on
the graph gives a solution.
If the graph is not Eulerian then some
retracing of streets is necessary and the
problem is more difficult.
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The Traveling Salesman Problem
A traveling salesman must visit n cities,
starting at one of the cities and returning
to it.
 If the distances between all cities is
known, what is the shortest possible
route?
 Google Search

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