§5.2 5.3 Graphs and Graph Terminology

§5.2 - 5.3 Graphs and Graph Terminology “Liesez Euler, Liesez Euler, c’est notre maître tous.” - Pierre Laplace Graphs consist of  points called vertices  lines called edges Edges connect two vertices Edges only intersect at vertices Edges joining a vertex to itself are called loops Example 1: The following picture is a graph List its vertices and A D edges C E B Example 2: This is also a graph The vertices just happen to have people’s names Such a graph could represent friendships (or any kind of relationship) Flexo Bender Leela Zoidberg QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture Fry Amy Farnsworth Now check out the graph below What can we say about it in comparison to the previous figure? Leela Fry Flexo Amy Bender QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture Farnsworth Zoidberg QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture Moral of the Story • One graph may be drawn in (infinitely) many ways, but it always provides us with the same information • Graphs are a structure for describing relationships between objects (The vertices denote the objects and the edges represent the relationship.) Graph Terminology Graph Terminology could ask e on on rg ja -y th ma e (ie - all th for) e r a s e c i t r e V t n e c a  Adj e r a t a h t s e c i t r e v two e g d e n a y b d e n i o j  Adjacent Edges are two edges that intersect at a vertex  The degree of a f o r e b m u n e h t s i x e t r ve edges at that vertex Graph Terminology e c i w t s t n u o c p o o l A toward the degree An vertex An vertex odd vertex is a of odd degree even vertex is a of even degree Example 3: 1) Find the degree of each vertex 2) Is A adjacent B? Is D adjacent A? Is E adjacent itself? Is C adjacent itself? A D to C to to E to 3) Is AB adjacent to BC? Is CE adjacent to BD? B Graph Terminology A path is a sequence t a h t h c u s s e c i t r e v of t n e c a j d a s i x e t r e each v to the next In a e b n a c e g d e h c a e , h t pa traveled only once h t a p a f o h t g n e l •The s e g d e f o r e b m u n e h t is in that path Example 3: (Exercise 60, pg 214) The map to the right of downtown Kingsburg, shows the Kings River running through the downtown area and the three islands (A, B, and C) connected to each other and both banks by seven bridges The Chamber of Commerce wants to design a walking tour that crosses all the bridges Draw a graph that models the layout of Kingsburg It was shown yesterday that it was possible to take a walk in such that you cross each bridge exactly once Show how N A B S C Slay- Example: • The Scooby Gang needs to patrol the following section of town starting at Sunnydale High (labeled G) Suppose that they must check each side of the street except for those along the park Find an optimal route for our intrepid demon hunters to take age Quiz 1, problem North Bank (N) B A C South Bank (S) Mathematics and the Arts? • One of Euler’s 800+ publications included a treatise on music theory • Book was too math-y for most composers too music-y for most mathematicians Mathematics and the Arts? • While Euler’s theories did not catch on, a relationship between mathematics and music composition does exist in what is called the golden ratio Fibonacci Numbers • The Fibonacci Numbers are those that comprise the sequence: 1, 1, 2, 3, 5, 8, 13, 21, • The sequence can be defined by: F1=1, F2=1; Fn=Fn-1+Fn-2 • These numbers can be used to draw a series of ‘golden’ rectangles like those to the right Fibonacci Numbers • The sequence of Fibonacci Ratios - fractions like 3/5, 5/8, 8/13 approach a number called the Golden Ratio (≈0.61803398…) The Golden Ratio • Several of Mozart’s piano sonatas make use of this ratio • At the time such pieces regularly employed a division into two parts Exposition and Development Recapitulation • In Piano Sonata No the change between parts occurs at measure 38 of 100 (which means that part is 62 ≈ 0.618 x 100) The Golden Ratio • Another example in music is in the ‘Hallelujah’ chorus in Handel’s Messiah • The piece is 94 measures long • Important events in piece: Entrance of trumpets “King of Kings” occurs in measures 57-58 ≈ (8/13) x 94 “The kingdom of glory…” occurs in meas 34-35 ≈ (8/13) x 57 etc, etc The Golden Ratio in Art H Approx = 0.618 x H The Golden Ratio in Art H Approx = 0.618 x H The Golden Ratio in Art The Golden Ratio in Art 618 x Ht 0.618 x Width The Golden Ratio in Art 618 x Ht 0.618 x Width ... information • Graphs are a structure for describing relationships between objects (The vertices denote the objects and the edges represent the relationship.) Graph Terminology Graph Terminology. .. bank and island as a vertex and each bridge as an edge joining them Euler was able to model the situation using the graph on the right Hence, the Königsberg puzzle is the same as asking if the graph. .. area and the three islands (A, B, and C) connected to each other and both banks by seven bridges The Chamber of Commerce wants to design a walking tour that crosses all the bridges Draw a graph