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

3. Lewinter & WidulskiThe Saga of Mathematics2 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.”

4. Lewinter & WidulskiThe Saga of Mathematics3 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.

5. Lewinter & WidulskiThe Saga of Mathematics4 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.

6. Lewinter & WidulskiThe Saga of Mathematics5 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.

7. Lewinter & WidulskiThe Saga of Mathematics6 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.

8. Lewinter & WidulskiThe Saga of Mathematics7 The 18 th 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 15 th 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 information age

9. Lewinter & WidulskiThe Saga of Mathematics8 The 18 th Century  The 18 th 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.

10. Lewinter & WidulskiThe Saga of Mathematics9 The 18 th 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.

11. Lewinter & WidulskiThe Saga of Mathematics10 The 18 th Century  The rulers of the 18 th 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.

12. Lewinter & WidulskiThe Saga of Mathematics11 The 18 th 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.

13. Lewinter & WidulskiThe Saga of Mathematics12 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.

14. Lewinter & WidulskiThe Saga of Mathematics13 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.

15. Lewinter & WidulskiThe Saga of Mathematics14 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.

16. Lewinter & WidulskiThe Saga of Mathematics15 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.

17. Lewinter & WidulskiThe Saga of Mathematics16 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

18. Lewinter & WidulskiThe Saga of Mathematics17 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!”

19. Lewinter & WidulskiThe Saga of Mathematics18 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.

20. Lewinter & WidulskiThe Saga of Mathematics19 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.Virgil’s Aeneid  Euler not only memorized the first 100 prime numbers but also their squares, cubes, fourth, fifth and sixth powers! prime numbers  He could also perform difficult calculations mentally, some of which required him to retain in his head 50 places of accuracy.

21. Lewinter & WidulskiThe Saga of Mathematics20 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.

22. Lewinter & WidulskiThe Saga of Mathematics21 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.

23. Lewinter & WidulskiThe Saga of Mathematics22 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.

24. Lewinter & WidulskiThe Saga of Mathematics23 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 e i  = cos  + i sin . e i  + 1 = 0

25. Lewinter & WidulskiThe Saga of Mathematics24 Euler’s Analysis Infinitorum  Euler showed that the general equation of second degree Ax 2 + Bxy + Cy 2 + 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 x 2 + y 2 = r 2 and a sphere centered at the origin is given by x 2 + y 2 + z 2 = r 2.

26. Lewinter & WidulskiThe Saga of Mathematics25 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.

27. Lewinter & WidulskiThe Saga of Mathematics26 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 x n + y n = z n is impossible for integers x, y, z, n =3 and n =4. (Fermat’s Last Theorem)

28. Lewinter & WidulskiThe Saga of Mathematics27 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.

29. Lewinter & WidulskiThe Saga of Mathematics28 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).

30. Lewinter & WidulskiThe Saga of Mathematics29 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.

31. Lewinter & WidulskiThe Saga of Mathematics30 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.

32. Lewinter & WidulskiThe Saga of Mathematics31 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.Eulerian Circles

33. Lewinter & WidulskiThe Saga of Mathematics32 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. 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

34. Lewinter & WidulskiThe Saga of Mathematics33 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

35. Lewinter & WidulskiThe Saga of Mathematics34 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.

36. Lewinter & WidulskiThe Saga of Mathematics35 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).

37. Lewinter & WidulskiThe Saga of Mathematics36 Leonhard Euler [1707-1783]  His publications account for one-third of all the technical articles published in 18 th 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?

38. Lewinter & WidulskiThe Saga of Mathematics37 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.

39. Lewinter & WidulskiThe Saga of Mathematics38 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.

40. Lewinter & WidulskiThe Saga of Mathematics39 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.

41. Lewinter & WidulskiThe Saga of Mathematics40 The Seven Bridges of Königsberg A B C D

42. Lewinter & WidulskiThe Saga of Mathematics41 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.

43. Lewinter & WidulskiThe Saga of Mathematics42 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.

44. Lewinter & WidulskiThe Saga of Mathematics43 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.

45. Lewinter & WidulskiThe Saga of Mathematics44 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.

46. Lewinter & WidulskiThe Saga of Mathematics45 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.”

47. Lewinter & WidulskiThe Saga of Mathematics46 §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.

48. Lewinter & WidulskiThe Saga of Mathematics47 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 Representation of a Simple Graph

49. Lewinter & WidulskiThe Saga of Mathematics48 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}} NH VT NY NJ MA RICT ME PA

50. Lewinter & WidulskiThe Saga of Mathematics49 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. NH VT NY NJ MA RICT ME PA

51. Lewinter & WidulskiThe Saga of Mathematics50 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

52. Lewinter & WidulskiThe Saga of Mathematics51 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.

53. Lewinter & WidulskiThe Saga of Mathematics52 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. Parallel edges

54. Lewinter & WidulskiThe Saga of Mathematics53 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...

55. Lewinter & WidulskiThe Saga of Mathematics54 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.

56. Lewinter & WidulskiThe Saga of Mathematics55 Types of Graphs: Summary  Keep in mind this terminology is not fully standardized... TermEdge TypeMultiple Edges ok? Self-loops ok? GraphUndir.No MultigraphUndir.YesNo PseudographUndirYes DigraphDirectedNoYes Directed Multigraph DirectedYes

57. Lewinter & WidulskiThe Saga of Mathematics56 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.

58. Lewinter & WidulskiThe Saga of Mathematics57 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.

59. Lewinter & WidulskiThe Saga of Mathematics58 Degree Sequence  If G is a graph with n nodes, the degree sequence (d 1, d 2, d 3, …, d n ) 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 P 5 or the graph G below. G P5P5

60. Lewinter & WidulskiThe Saga of Mathematics59 §2: Graph Theory Concepts  The graph G below will be used to demonstrate several concepts in graph theory. a b c d e f g h i j G

61. Lewinter & WidulskiThe Saga of Mathematics60 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 e f g h i j G

62. Lewinter & WidulskiThe Saga of Mathematics61 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.

63. Lewinter & WidulskiThe Saga of Mathematics62 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}

64. Lewinter & WidulskiThe Saga of Mathematics63 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.

65. Lewinter & WidulskiThe Saga of Mathematics64 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 

66. Lewinter & WidulskiThe Saga of Mathematics65 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.

67. Lewinter & WidulskiThe Saga of Mathematics66 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

68. Lewinter & WidulskiThe Saga of Mathematics67 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.

69. Lewinter & WidulskiThe Saga of Mathematics68 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.

70. Lewinter & WidulskiThe Saga of Mathematics69 §3: Special Classes of Graphs  Complete graphs K n  Cycles C n  Regular Graphs  Paths P n  Wheels W n  Hypercubes or n-Cubes Q n  Bipartite graphs  Complete bipartite graphs K m,n  The n-dimensional Octahedron

71. Lewinter & WidulskiThe Saga of Mathematics70 Complete Graphs  For any positive integer n, a complete graph on n vertices, K n, is a graph with n nodes in which every node is adjacent to every other node. K1K1 K2K2 K3K3 K4K4 K5K5 K6K6 Note: K n has edges.

72. Lewinter & WidulskiThe Saga of Mathematics71 Cycles  For any n3, a cycle on n vertices, C n, is a graph where V={v 1,v 2,…,v n } and E={{v 1,v 2 },{v 2,v 3 },…,{v n1,v n },{v n,v 1 }}. C3C3 C4C4 C5C5 C6C6 C7C7 C8C8 How many edges are there in C n ?

73. Lewinter & WidulskiThe Saga of Mathematics72 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 C n are 2- regular.  The complete graphs K n are (n–1)-regular. Can you draw a 3-regular graph on six nodes?

74. Lewinter & WidulskiThe Saga of Mathematics73 Paths  Another very important class of graphs are paths, denoted P n, where n is, once again, the number of vertices in the path. P 5. P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 How many edges are there in P n ?

75. Lewinter & WidulskiThe Saga of Mathematics74 Wheels  For any n3, a wheel W n, is a graph obtained by taking the cycle C n-1 and adding one extra vertex v hub and n-1 extra edges {{v hub,v 1 }, {v hub,v 2 },…,{v hub,v n-1 }}. How many edges are there in W n ?

76. Lewinter & WidulskiThe Saga of Mathematics75 Hypercubes (n-cubes)  For any positive integer n, the hypercube Q n is a simple graph consisting of two copies of Q n-1 connected together at corresponding nodes. Q 0 has 1 node. Number of vertices: 2 n. Number of edges: Exercise to try! Q3Q3 Q0Q0 Q1Q1 Q2Q2 Q4Q4

77. Lewinter & WidulskiThe Saga of Mathematics76 Bipartite Graphs  A bipartite graph G is a graph whose vertex set can be partitioned into two subsets V 1 and V 2 such that every edge of G joins V 1 with V 2. Q3Q3 Q3Q3 Q3Q3 Q3Q3 Theorem: A graph is bipartite iff all its cycles are even.

78. Lewinter & WidulskiThe Saga of Mathematics77 Complete Bipartite Graphs  A complete bipartite graph, K m,n, is a bipartite graph which contains every edge joining V 1 and V 2. K 2,3 K 3,3 K 4,4

79. Lewinter & WidulskiThe Saga of Mathematics78 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

80. Lewinter & WidulskiThe Saga of Mathematics79 §4: Graph Operations  Subgraphs  Unions  Complement  Join (omitted)  Product (omitted)  Composition (omitted)

81. Lewinter & WidulskiThe Saga of Mathematics80 Subgraphs  A subgraph of a graph G=(V,E) is a graph H=(W,F) where WV and FE. GH

82. Lewinter & WidulskiThe Saga of Mathematics81 Subgraph Example  The hypercube Q 3 is a subgraph of the complete bipartite K 4,4. K 4,4 Q3Q3

83. Lewinter & WidulskiThe Saga of Mathematics82 Graph Unions  The union G 1 G 2 of two simple graphs G 1 =(V 1, E 1 ) and G 2 =(V 2,E 2 ) is the simple graph (V 1 V 2, E 1 E 2 ). G1G1 G2G2 G1G2G1G2

84. Lewinter & WidulskiThe Saga of Mathematics83 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. GG

85. Lewinter & WidulskiThe Saga of Mathematics84 §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.

86. Lewinter & WidulskiThe Saga of Mathematics85 Adjacency Lists  A table with 1 row per vertex, listing its adjacent vertices. VertexAdjacent Vertices ab, f ba, d, f cd db, c, f, e fa, b, d a b d c f e

87. Lewinter & WidulskiThe Saga of Mathematics86 Directed Adjacency Lists  1 row per node, listing the terminal nodes of each edge incident from that node. VertexAdjacent Vertices ab, f bd c dc e fb, d a b d c f e

88. Lewinter & WidulskiThe Saga of Mathematics87 Adjacency Matrix  Matrix A=[a ij ], where a ij is 1 if {v i, v j } is an edge of G, 0 otherwise. a b d c f e abcdef a 010001 b 100101 c 000100 d 011001 e 000000 f 110100

89. Lewinter & WidulskiThe Saga of Mathematics88 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.

90. Lewinter & WidulskiThe Saga of Mathematics89 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 4 5 12345 a 10010 b 11001 c 01100 d 00111

91. Lewinter & WidulskiThe Saga of Mathematics90 Graph Isomorphism  Formal definition: Simple graphs G 1 =(V 1, E 1 ) and G 2 =(V 2, E 2 ) are isomorphic if and only if there exists a bijection f:V 1 V 2 such that for all a,b  V 1, a and b are adjacent in G 1 if and only if f(a) and f(b) are adjacent in G 2. f is the “renaming” function that makes the two graphs identical. Definition can easily be extended to other types of graphs.

92. Lewinter & WidulskiThe Saga of Mathematics91 Graph Invariants under Isomorphism  Necessary but not sufficient conditions for G 1 =(V 1,E 1 ) to be isomorphic to G 2 =(V 2,E 2 ): |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.

93. Lewinter & WidulskiThe Saga of Mathematics92 Isomorphism Example  If isomorphic, label the 2nd graph to show the isomorphism, else identify difference. a b c d e f b d a e f c

94. Lewinter & WidulskiThe Saga of Mathematics93 Are These Isomorphic?  If isomorphic, label the 2nd graph to show the isomorphism, else identify difference. a b c d e * Same # of nodes * Same # of edges * Different # of nodes of degree 2! (1 versus 3)

95. Lewinter & WidulskiThe Saga of Mathematics94 Self Complementary Graphs  The self-complementary graph is isomorphic with its complement.  Show that P 4 is self-complementary. GG = ~

96. Lewinter & WidulskiThe Saga of Mathematics95 §6: Walks, Trials, and Paths  A walk of a graph G is an alternating sequence of nodes and edges v 0, e 1, v 1, e 2, v 2, e 3, v 3, …, v n-1, e n, v n  beginning and ending with nodes, such that each edge is incident with the two nodes immediately preceding and following it.  This walk, called a v 0 -v n walk, joins v 0 and v n and may also be denoted v 0, v 1, v 2, v 3,…, v n-1, v n.

97. Lewinter & WidulskiThe Saga of Mathematics96 Walks, Trials, and Paths  It is a closed walk if v 0 =v n, 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.

98. Lewinter & WidulskiThe Saga of Mathematics97 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 e f g h i j G

99. Lewinter & WidulskiThe Saga of Mathematics98 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 e f g h i j G

100. Lewinter & WidulskiThe Saga of Mathematics99 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.

101. Lewinter & WidulskiThe Saga of Mathematics100 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 C 3, with vertices e, f and g, and C 4, with vertices c, d, g and h.

102. Lewinter & WidulskiThe Saga of Mathematics101 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. a b d c f e

103. Lewinter & WidulskiThe Saga of Mathematics102 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.

104. Lewinter & WidulskiThe Saga of Mathematics103 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. v1v1 v2v2 v3v3 v4v4

105. Lewinter & WidulskiThe Saga of Mathematics104 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.

106. Lewinter & WidulskiThe Saga of Mathematics105 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 K n cannot be disconnected by removing any number of nodes, but the trivial graph results after removing n – 1 nodes; thus, κ(K n ) = n – 1.

107. Lewinter & WidulskiThe Saga of Mathematics106 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'(K 1 ) = 0, and the edge-connectivity of a disconnected graph is 0, while the connectivity of a graph with a bridge is 1. κ'(K n ) = n – 1.

108. Lewinter & WidulskiThe Saga of Mathematics107 §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 K 4 is planar.

109. Lewinter & WidulskiThe Saga of Mathematics108 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 K 5 planar?  The answer is NO! x y K 5 minus an edge is planar.

110. Lewinter & WidulskiThe Saga of Mathematics109 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 WG BC E

111. Lewinter & WidulskiThe Saga of Mathematics110 Water, Gas, and Electricity  This is equivalent to asking if the graph K 3,3 is planar.  The answer is NO!  Again this is almost true, but not quite.  If we remove a single edge from K 3,3 it becomes planar, but however we try to draw the last edge it will cross another edge.  Therefore, both K 5 and K 3,3 are not planar.

112. Lewinter & WidulskiThe Saga of Mathematics111 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.

113. Lewinter & WidulskiThe Saga of Mathematics112 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

114. Lewinter & WidulskiThe Saga of Mathematics113 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.

115. Lewinter & WidulskiThe Saga of Mathematics114 Euler Formula Example 1  For the graph K 4, V = 4 E = 6 F = 4  So V – E + F = 2. 1 2 3 4 “the outside”

116. Lewinter & WidulskiThe Saga of Mathematics115 Euler’s Formula Example 2  Show V – E + F = 2 for the dodecahedron.

117. Lewinter & WidulskiThe Saga of Mathematics116 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.

118. Lewinter & WidulskiThe Saga of Mathematics117 Example 1  The complete graph K 5 is not planar. Notice that for this graph, V = 5 and E = 10. Each cycle of K 5 contains at least 3 edges. Since V – E + F = 2, implies F = 7 if K 5 is planar. By the inequality kF ≤ 2E. 21 = 3F ≤ 2E = 20. Contradiction!

119. Lewinter & WidulskiThe Saga of Mathematics118 Example 2  The complete bipartite graph K 3,3 is not planar. Notice that V = 6 and E = 9. So using Euler’s formula V – E + F = 2, implies F = 5 if K 3,3 is planar. Each cycle of K 3,3 contains at least 4 edges. By the inequality kF ≤ 2E. 20 = 4F ≤ 2E = 18. Contradiction! K 3,3

120. Lewinter & WidulskiThe Saga of Mathematics119 Kuratowski’s Theorem  If G is a finite graph, then the following conditions are equivalent: G is not planar. G contains a homeomorph of K 5 or K 3,3.  A homeomorph means that the nodes of the graph are identified with the nodes of K 5 or K 3,3 and the edges are identified with disjoint paths.

121. Lewinter & WidulskiThe Saga of Mathematics120 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.

122. Lewinter & WidulskiThe Saga of Mathematics121 §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?

123. Lewinter & WidulskiThe Saga of Mathematics122 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.

124. Lewinter & WidulskiThe Saga of Mathematics123 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?

125. Lewinter & WidulskiThe Saga of Mathematics124 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.

126. Lewinter & WidulskiThe Saga of Mathematics125 The 3-dimensional Octahedron  The 3-dimensional Octahedron is Eulerian.

127. Lewinter & WidulskiThe Saga of Mathematics126 Other Examples  The complete graph K n is Eulerian if and only if n is odd (because the degree of each node of K n 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).

128. Lewinter & WidulskiThe Saga of Mathematics127 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 WWW links

129. Lewinter & WidulskiThe Saga of Mathematics128 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.

130. Lewinter & WidulskiThe Saga of Mathematics129 Round-the-World Puzzle  Can we traverse all the vertices of a dodecahedron, visiting each once? Dodecahedron Puzzle Equivalent Graph Pegboard Version

131. Lewinter & WidulskiThe Saga of Mathematics130 The 3-dimensional Octahedron  The 3-dimensional Octahedron is Hamiltonian.

132. Lewinter & WidulskiThe Saga of Mathematics131 Other Examples  The complete graph K n 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).

133. Lewinter & WidulskiThe Saga of Mathematics132 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.

134. Lewinter & WidulskiThe Saga of Mathematics133 The Two-Way Street Problem This is clearly Eulerian, since each node has even degree.

135. Lewinter & WidulskiThe Saga of Mathematics134 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.

136. Lewinter & WidulskiThe Saga of Mathematics135 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 Google Search