Lewinter & WidulskiThe Saga of Mathematics1 The Age of Euler Chapter 10 Part 1.

Lewinter & Widulski The Saga of Mathematics 1

The Age of Euler

Chapter 10Part 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.”


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.

A Famous Tale


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,

donc Dieu existe; répondez!” That is, “Sir, , hence God exists; reply!”

xn

ba n

xn

ba n


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.


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.


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.


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.


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.


Euler’s Analysis Infinitorum 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 .

ei + 1 = 0


Euler’s Analysis Infinitorum Euler showed that the general equation of

second degreeAx2 + 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.


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.


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)


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.


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).


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.


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.


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.


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 Fermat-

Euler theorem). Euler’s pentagonal formula for partitions. Eulerian graphs


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


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.


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).


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?


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.


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.


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.


The Seven Bridges of Königsberg

A

B

C

D


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.


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.


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.


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.


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.”


§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.


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.

Visual Representationof a Simple Graph


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}}

NHVT

NY

NJ

MA

RICT

ME

PA


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.

NHVT

NY

NJ

MA

RICT

ME

PA


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


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.


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.

Paralleledges


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:EVV.

E.g., V=web pages,E=hyperlinks. The WWW isa directed multigraph...


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,vV}. Edge eE is a loop if f(e)={u,u}={u}.

E.g., nodes are campsitesin a state park, edges arehiking trails through the woods.


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


Adjacency Let G be a graph with edge set E. Let eE 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.


Degree of a Vertex Let G be a graph and vV 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.


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.

GP5


§2: Graph Theory Concepts The graph G below will be used to

demonstrate several concepts in graph theory.

a

b

c d

ef

g hi

jG


Degree of a Vertex The degree of a vertex is the number of

edges touching it (technically, incident with it).

Thus, the degree of vertex g in graph G above is 4. This is written as deg(g)=4.

a

b

c d

ef

g hi

jG


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.


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}


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.


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

Ev

Vv

2)deg(


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.


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


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.


Directed Handshaking Theorem Let G be a digraph with vertex set V and

edge set E. Then:

Note that the degree of a node is unchanged by whether we consider its edges to be directed or undirected.

EvvvVvVvVv

)deg(2

1)(deg)(deg


§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


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 K2K3

K4 K5 K6

Note: Kn has edges.2

)1(1

1

nni

n

i


Cycles For any n3, a cycle on n vertices, Cn, is a

graph where V={v1,v2,… ,vn} and E={{v1,v2},{v2,v3},…,{vn1,vn},{vn,v1}}.

C3 C4 C5 C6 C7C8

How many edges are there in Cn?


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 2-regular.

The complete graphs Kn are (n–1)-regular. Can you draw a 3-regular graph on six nodes?


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?


Wheels For any n3, 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 W8W9

How many edges are there in Wn?


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.

Number of vertices: 2n. Number of edges: Exercise to try!

Q3

Q0Q1 Q2 Q4


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 Q3Q3 Q3

Theorem: A graph is bipartite iff all its cycles are even.


Complete Bipartite Graphs A complete bipartite graph, Km,n, is a

bipartite graph which contains every edge joining V1 and V2.

K2,3 K3,3 K4,4


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


§4: Graph Operations Subgraphs Unions Complement Join (omitted) Product (omitted) Composition (omitted)


Subgraphs A subgraph of a graph G=(V,E) is a graph

H=(W,F) where WV and FE.

G H


Subgraph Example The hypercube Q3 is a subgraph of the

complete bipartite K4,4.

K4,4Q3


Graph Unions The union G1G2 of two simple graphs

G1=(V1, E1) and G2=(V2,E2) is the simple graph (V1V2, E1E2).

G1

G2 G1G2


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 G


§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.


Adjacency Lists A table with 1 row per

vertex, listing its adjacent vertices. Vertex Adjacent Vertices

a b, f

b a, d, f

c d

d b, c, f,

e

f a, b, d

a

b

d

c

f

e


Directed Adjacency Lists 1 row per node, listing

the terminal nodes of each edge incident from that node.

Vertex Adjacent Vertices

a b, f

b d

c

d c

e

f b, d

a

b

d

c

f

e


Adjacency Matrix Matrix A=[aij], where

aij is 1 if {vi, vj} is an edge of G, 0 otherwise.

a

b

d

c

f

e

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


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.


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. First defined by the

physicist Kirchhoff (1847).

Each column contains exactly two ones. Why?

a

bd

c

1

23

45

1 2 3 4 5

a 1 0 0 1 0

b 1 1 0 0 1

c 0 1 1 0 0

d 0 0 1 1 1


Graph Isomorphism Formal definition:

Simple graphs G1=(V1, E1) and G2=(V2, E2) are isomorphic if and only if there exists a bijection f:V1V2 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.


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.


Isomorphism Example If isomorphic, label the 2nd graph to show

the isomorphism, else identify difference.

a

b

cd

ef

b

d

a

efc


Are These Isomorphic? If isomorphic, label the 2nd graph to show

the isomorphism, else identify difference.

ab

c

d

e

* Same # ofnodes

* Same # ofedges

* Different# of nodes of

degree 2! (1 versus 3)


Self Complementary Graphs The self-complementary graph is

isomorphic with its complement.

Show that P4 is self-complementary.

G G

=~


§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.


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.


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

b

c d

ef

g hi

jG


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

b

c d

ef

g hi

jG


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.


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.


Paths in Directed Graphs Same as in undirected graphs, but the

path must go in the direction of the arrows.

In the digraph to the right abdc is apath.

a

b

d

c

f

e


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.


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 v2 v3

v4


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 vice-versa.


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.


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.


§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.


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!

x

y K5 minus an edge is

planar.


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?

A

W G

B C

E


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.


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.


Euler’s Formula For any convex polyhedron,

V – E + F = 2 V = Vertices E = Edges F = Faces

Examples 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


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.


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”


Euler’s Formula Example 2 Show V – E + F = 2 for the dodecahedron.


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.


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!


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!

K3,3


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.


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.


§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?


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.


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?


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.


The 3-dimensional Octahedron The 3-dimensional Octahedron is Eulerian.


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).


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


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.


Round-the-World Puzzle Can we traverse all the vertices of a

dodecahedron, visiting each once?

DodecahedronPuzzle

EquivalentGraph

PegboardVersion


The 3-dimensional Octahedron The 3-dimensional Octahedron is

Hamiltonian.


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).


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.


The Two-Way Street Problem

This is clearly Eulerian, since each node has even degree.


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.


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?

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Lewinter & WidulskiThe Saga of Mathematics1 The Age of Euler Chapter 10 Part 1.

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