1. Week 11 - Wednesday

2.  What did we talk about last time?  Graphs  Euler paths and tours

4.  One hundred ants are walking along a meter long stick  Each ant is traveling either to the left or the right with a constant speed of 1 meter per minute  When two ants meet, they bounce off each other and reverse direction  When an ant reaches an end of the stick, it falls off  Will all the ants fall off?  What is the longest amount of time that you would need to wait to guarantee that all ants have fallen off?

6.  An Euler circuit has to visit every edge of a graph exactly once  A Hamiltonian circuit must visit every vertex of a graph exactly once (except for the first and the last)  If a graph G has a Hamiltonian circuit, then G has a subgraph H with the following properties:  H contains every vertex of G  H is connected  H has the same number of edges as vertices  Every vertex of H has degree 2  In some cases, you can use these properties to show that a graph does not have a Hamiltonian circuit  In general, showing that a graph has or does not have a Hamiltonian circuit is NP-complete (widely believed to take exponential time)  Does the following graph have a Hamiltonian circuit? e b ac d

8.  As you presumably know, a matrix is a rectangular array of elements  An m x n matrix has m rows and n columns

9.  There are many, many different ways to represent a graph  If you get tired of drawing pictures or listing ordered pairs, a matrix is not a bad way  To represent a graph as an adjacency matrix, make an n x n matrix, where n is the number of vertices  Let the nonnegative integer at a ij give the number of edges from vertex i to vertex j  A simple graph will always have either 1 or 0 for every location

10.  What is the adjacency matrix for the following graph?  What about for this one? v1v1 v1v1 v3v3 v3v3 v2v2 v2v2 v3v3 v3v3 v2v2 v2v2 v1v1 v1v1

11.  Draw a graph corresponding to this matrix

12.  What's the adjacency matrix of this graph?  Note that the matrix is symmetric  In a symmetric matrix, a ij = a ji for all 1 ≤ i ≤ n and 1 ≤ j ≤ n  All undirected graphs have a symmetric matrix representation v1v1 v1v1 v4v4 v4v4 v2v2 v2v2 v3v3 v3v3

13.  To multiply matrices A and B, it must be the case that A is an m x k matrix and that B is a k x n matrix  Then, the i th row, j th column of the result is the dot product of the i th row of A with the j th column of B  In other words, we could compute element c ij in the result matrix C as follows:

14.  Multiply matrices A and B

15.  Matrix multiplication is associative  That is, A(BC) = (AB)C  Matrix multiplication is not commutative  AB is not always equal to BA (for one thing, BA might not even be legal if AB is)  There is an n x n identity matrix I such that, for any m x n matrix A, AI = A  I is all zeroes, except for the diagonal (where row = column) which is all ones  We can raise square matrices to powers using the following recursive definition  A 0 = I, where I is the n x n identity matrix  A k = AA k-1, for all integers k ≥ 1

16.  Is A symmetric?  Compute A 0  Compute A 1  Compute A 2  Compute A 3

17.  We can find the number of walks of length k that connect two vertices in a graph by raising the adjacency matrix of the graph to the k th power  Raising a matrix to the zeroth power means you can only get from a vertex to itself (identity matrix)  Raising a matrix to the first power means that the number of paths of length one from one vertex to another is exactly the number of edges between them  The result holds for all k, but we aren't going to prove it

19.  A property is called an isomorphism invariant if its truth or falsehood does not change when considering a different (but isomorphic) graph  10 common isomorphism invariants: 1. Has n vertices 2. Has m edges 3. Has a vertex of degree k 4. Has m vertices of degree k 5. Has a circuit of length k 6. Has a simple circuit of length k 7. Has m simple circuits of length k 8. Is connected 9. Has an Euler circuit 10. Has a Hamiltonian circuit

20.  If any of the invariants have different values for two different graphs, those graphs are not isomorphic  Use the 10 invariants given to show that the following pair of graphs is not isomorphic

21. Student Lecture

23.  A tree is a graph that is circuit-free and connected  Examples: A graph made up of disconnected trees is called a forest

24.  Trees have almost unlimited applications  You should all be familiar with the concept of a decision tree from programming Score on Part I Score on Part II Math 120 Math 110 Math 100 < 8 = 8, 9, 10 > 10  10 > 6  6 6

25.  A grammar for a formal language (such as we will discuss next week or the week after) is made up of rules that allow non- terminals to be turned into other non-terminals or terminals  For example: 1.  2.  | 3.  4.  a | an | the 5.  funky 6.  DJ | beat 7.  plays | spins  Make a parse tree corresponding to the sentence, "The DJ plays a funky beat"

26.  Any tree with more than one vertex has at least one vertex of degree 1  If a vertex in a tree has degree 1 it is called a terminal vertex (or leaf)  All vertices of degree greater than 1 in a tree are called internal vertices (or branch vertices)

27.  For any positive integer n, a tree with n vertices must have n – 1 edges  Prove this by mathematical induction  Hint: Any tree with 2 or more nodes has a vertex of degree 1. What happens when you remove that vertex?

28.  In a rooted tree, one vertex is distinguished and called the root  The level of a vertex is the number of edges along the unique path between it and the root  The height of a rooted tree is the maximum level of any vertex of the tree  The children of any vertex v in a rooted tree are all those nodes that are adjacent to v and one level further away from the root than v  Two distinct vertices that are children of the same parent are called siblings  If w is a child of v, then v is the parent of w  If v is on the unique path from w to the root, then v is an ancestor of w and w is a descendant of v

29.  Consider the following tree, rooted at 0  What is the level of 5?  What is the level of 0?  What is the height of this tree?  What are the children of 3?  What is the parent of 2?  What are the siblings of 8?  What are the descendants of 3? 3 3 0 0 5 5 2 2 1 1 7 7 9 9 6 6 4 4 8 8

30.  A binary tree is a rooted tree in which every parent has at most two children  Each child is designated either the left child or the right child  In a full binary tree, each parent has exactly two children  Given a parent v in a binary tree, the left subtree of v is the binary tree whose root is the left child of v  Ditto for right subtree

31.  As we all know from data structures, a binary tree can be used to make a data structure that is efficient for insertions, deletions, and searches  But, it doesn't stop there!  We can represent binary arithmetic with a binary tree  Make a binary tree for the expression ((a – b)∙c) + (d/e)  The root of each subtree is an operator  Each subtree is either a single operand or another expression

32.  If k is a positive integer and T is a full binary tree with k internal vertices, then T has a total 2k + 1 vertices and has k + 1 terminal vertices  Prove it!  Hint: Induction isn't needed. We just need to relate the number of non-terminal nodes to the number of terminal nodes

33.  If T is a full binary tree with height h, then it has 2 h+1 – 1 vertices  Prove it using induction!

34.  If T is a binary tree with t terminal vertices and height h, then t  2 h  Prove it using strong induction on the height of the tree  Hint: Consider cases where the root of the tree has 1 child and 2 children separately

37.  Spanning trees  Graphing functions

38.  Keep reading Chapter 10  Start Chapter 11  Start work on Assignment 9  Due next Friday