The Age of Euler

1
The Age of Euler

• Chapter 10
• Part 1

2
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.

3
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.

4
Leonhard Euler 1707-1783

• At the age of 15, he received his Bachelors
  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, Jacobs brother, Euler switched to
  mathematics.

5
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.

6
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.

7
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.

8
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.

9
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.

10
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 ones position
  while out at sea.
• This position is determined by observing the
  heavens.

11
The 18th Century

• After Newtons 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.

12
Leonhard Euler 1707-1783

• Euler eventually obtained royal appointments in
  several European courts including Russia and
  Germany (under Frederick the Great).
• Two of Eulers 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.

13
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 Eulers
  work.

14
Leonhard Euler 1707-1783

• Upon Nicholas Bernoullis 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.

15
Eulers Mechanica (1736)

• First textbook in which Newtons 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.

16
Euler and the Atheist
A Famous Tale

• 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.

17
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!

18
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, Eulers argument was nonsense but
  Diderot didnt see it.
• Euler would eventually meet his match in
  arguments with Voltaire.

19
Leonhard Euler 1707-1783

• Euler had a phenomenal memory.
• As a boy, Euler memorized Virgils 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.

20
Leonhard Euler 1707-1783

• Eulers 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.

21
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.

22
Eulers 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.

23
Eulers 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 ?.

24
Eulers 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.

25
Eulers Institutiones calculi integralis

• A thorough investigation of integrals.
• It includes Taylors 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.

26
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, n3 and n4.
  (Fermats Last Theorem)

27
Leonhard Euler 1707-1783

• In 1744 appeared Eulers 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.

28
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).

29
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.

30
Eulers Letters to a German Princess

• During Eulers 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.

31
Eulers Letters to a German Princess

• They became immensely popular and were circulated
  in seven languages.
• William Dunham says the they are one of historys
  finest example of popular science.
• What we call Venn diagrams first appears in
  Eulers Letters.
• Venn himself first called them "Eulerian
  Circles", but then somehow managed to get them
  called Venn Diagrams.

32
Leonhard Euler 1707-1783

• Many other results of Euler can be found in his
  smaller papers.
• Some of the better known results are
• Eulers Polyhedron Formula V E F 2.
• The Euler Line of a Triangle.
• Eulers constant ? 0.577215664901532.
• Euler's theorem (also known as the Fermat-Euler
  theorem).
• Eulers pentagonal formula for partitions.
• Eulerian graphs

33
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

34
Leonhard Euler 1707-1783

• To get an idea of the magnitude of Eulers work
  it is worth noting that
• Euler wrote more than 500 books and papers during
  his lifetime about 800 pages per year.
• After Eulers death, it took over forty years for
  the backlog of his work to appear in print.
• Approximately 400 more publications.

35
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).

36
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?

37
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.

38
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.

39
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.

40
The Seven Bridges of Königsberg
A
D
B
C
41
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.

42
The Bridges of Königsberg Problem

• Lets 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.

43
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.

44
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
  Leibnizs desire for a type of geometry that
  doesnt 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.

45
Eulers 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.

46
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.

47
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.

48
Example of a Graph

• Let V be the set of states in the north eastern
  part of the U.S.
• VME, NH, VT, MA, RI, CT, NY, NJ, PA
• Let Eu,vu 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

49
Example of a Graph (continued)

• The specific layout, or representation, of the
  graph doesnt 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.

50
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
51
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.

52
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.

53
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 fE?V?V.
• E.g., Vweb pages,Ehyperlinks. The WWW isa
  directed multigraph...

54
Pseudographs

• Like a multigraph, but edges connecting a node to
  itself are allowed.
• A pseudograph G(V, E, f ) wherefE?u,vu,v?V
  . Edge e?E is a loop if f(e)u,uu.
• E.g., nodes are campsitesin a state park, edges
  arehiking trails through the woods.

55
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
56
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.

57
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.

58
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.

59
2 Graph Theory Concepts

• The graph G below will be used to demonstrate
  several concepts in graph theory.

60
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.

61
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.

62
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

63
Vertex Set and Edge Set

• In this case, graph G has ten vertices, so n10.
• 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.

64
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

65
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.

66
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

67
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 vs in-degree and out-degree.

68
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.

69
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

70
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.

Note Kn has edges.
71
Cycles

• For any n?3, a cycle on n vertices, Cn, is a
  graph where Vv1,v2, ,vn and
  Ev1,v2,v2,v3,,vn?1,vn,vn,v1.

How many edges are there in Cn?
72
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 (n1)-regular.
• Can you draw a 3-regular graph on six nodes?

73
Paths

• Another very important class of graphs are paths,
  denoted Pn, where n is, once again, the number of
  vertices in the path. P5.

How many edges are there in Pn?
74
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.

How many edges are there in Wn?
75
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!
76
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.

Theorem A graph is bipartite iff all its cycles
are even.
77
Complete Bipartite Graphs

• A complete bipartite graph, Km,n, is a bipartite
  graph which contains every edge joining V1 and V2.

78
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.

79
4 Graph Operations

• Subgraphs
• Unions
• Complement
• Join (omitted)
• Product (omitted)
• Composition (omitted)

80
Subgraphs

• A subgraph of a graph G(V,E) is a graph H(W,F)
  where W?V and F?E.

81
Subgraph Example

• The hypercube Q3 is a subgraph of the complete
  bipartite K4,4.

82
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
G1?G2
83
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.

84
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.

85
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
86
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
87
Adjacency Matrix

• Matrix Aaij, 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
88
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.

89
Incidence Matrix
a

• 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?

1
4
5
b
d
2
3
c
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
90
Graph Isomorphism

• Formal definition
• Simple graphs G1(V1, E1) and G2(V2, E2) are
  isomorphic if and only if there exists a
  bijection fV1?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.

91
Graph Invariants under Isomorphism

• Necessary but not sufficient conditions for
  G1(V1,E1) to be isomorphic to G2(V2,E2)
• V1V2, E1E2.
• 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.

92
Isomorphism Example

• If isomorphic, label the 2nd graph to show the
  isomorphism, else identify difference.

d
b
a
e
f
c
93
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)
94
Self Complementary Graphs

• The self-complementary graph is isomorphic with
  its complement.
• Show that P4 is self-complementary.

95
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.

96
Walks, Trials, and Paths

• It is a closed walk if v0vn, 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, n3, is a cycle.
• The length of a walk, trail or path is the number
  of edges that occur in it.

97
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.

98
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.

99
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.

100
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.

101
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 a
• path.

102
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.

103
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.

v4
v3
v1
v2
104
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.

105
Connectivity

• The connectivity ? ?(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,
  ?(Kn) n 1.

106
Edge-Connectivity

• The edge-connectivity ?' ?'(G) of a graph G is
  the minimum number of edges whose removal results
  in a disconnected or trivial graph.
• Thus ?'(K1) 0, and the edge-connectivity of a
  disconnected graph is 0, while the connectivity
  of a graph with a bridge is 1.
• ?'(Kn) n 1.

107
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.

108
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!

K5 minus an edge is planar.
109
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
B
C
W
G
E
110
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.

111
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.

112
Eulers Formula

• For any convex polyhedron,
• V E F 2
• V Vertices
• E Edges
• F Faces
• Examples
• Tetrahedron V4, E6, F4
• Cube V8, E12, F6
• Octahedron V6, E12, F8
• Dodecahedron V20, E30, F12
• Icosahedron V12, E30, F20
• BuckyBall V60, E90, F32

113
Proof of Eulers Formula

• Proof by induction
• If no edges, its an isolated vertex. So V1, E0,
  F1
• 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.

114
Euler Formula Example 1

• For the graph K4,
• V 4
• E 6
• F 4
• So V E F 2.

115
Eulers Formula Example 2

• Show V E F 2 for the dodecahedron.

116
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.

117
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!

118
Example 2

• The complete bipartite graph K3,3 is not planar.
• Notice that V 6 and E 9.
• So using Eulers 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!

119
Kuratowskis 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.

120
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.
121
8 Traversability

• Eulers 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?

122
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.

123
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?
124
Fleurys 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.

125
The 3-dimensional Octahedron

• The 3-dimensional Octahedron is Eulerian.

126
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).

127
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

128
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.

129
Round-the-World Puzzle

• Can we traverse all the vertices of a
  dodecahedron, visiting each once?

EquivalentGraph
Pegboard Version
Dodecahedron Puzzle
130
The 3-dimensional Octahedron

• The 3-dimensional Octahedron is Hamiltonian.

131
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).

132
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.

133
The Two-Way Street Problem
This is clearly Eulerian, since each node has
even degree.
134
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.

135
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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