Week 12: Trees; Review. 22 and 24 November, 2017

Transcription

1 (1/24) MA284 : Discrete Mathematics Week 12: Trees; Review 22 and 24 November, 2017 C C C C 1 Trees Recall... Applications: Chemistry Applications: Decision Trees Spanning trees 2 What we didn t study Directed graphs Adjacency Matrix 3 Review The Final Exam Past exam papers Revision Questions 4 A summary in one slide (1/2)...

2 Reminder... (2/24) The deadline for ASSIGNMENT 4 is 5pm, Friday, 24 November To access the assignment, go to link! There are 10 questions. You may attempt each one up to 10 times. This assignment contributes 5% to your final grade for Discrete Mathematics. For more information, see Blackboard, or link!

3 Trees Recall... (3/24) Path: A PAT in a graph is a sequence of adjacent vertices in a graph. Connected: A graph is CONNECTED if there is a path between every pair of vertices.n Circuit: A CIRCUIT in a graph is a path that starts and ends at the same vertex, and no edge is repeated. Cycle: A CYCLE is a PAT that starts and ends at the same vertex, but visits no other vertex twice. Acyclic: A graph that has no circuits is called ACYCLIC. Tree: A TREE is a connected, acyclic graph. Last week we learned that if a tree has e edges, and v vertices, then e = v 1. Conversely, if a graph with v vertices has no cycles/circuits, and has e = v 1 edges, then it is a tree.

4 Trees Applications: Chemistry (4/24) There are many, many applications, of trees in mathematics, computer science, and the applied sciences. As already mentioned, the mathematical study of trees began in Chemistry. Example Saturated hydrocarbons isomers (alkane) are of the form C n 2n+2. They have n carbon atoms, and 2n + 2 hydrogen atoms. The carbon atoms can bond with 4 other atoms, and the hydrogens with just one. Show that the graph of all such isomers are trees.

5 Trees Applications: Chemistry (5/24) C C C C C C C C C C C C C C C C

6 Trees Applications: Decision Trees (6/24) A DECISION TREE is a graph where each node represents a possibility, and each branch/edge from that node is a possible outcome. Example (Q1, MA204 exam, 2014/2015) General Incompetence and Major Disaster played a chess match in which there were no drawn games. The first player to win three games in a row or a total of four games won the match. The General won her first game and the person who won the second game also won the third game. Construct an appropriate tree diagram to find the number of ways in which the match may have proceeded.

7 Trees Applications: Decision Trees (7/24) Puzzle You have eight identical-looking coins, but one is a counterfeit and lighter than the rest. You have a balance scale. Show that you can find the counterfeit one with just two weighing. ow many weighing are needed for nine coins? And ten?

8 Trees Spanning trees (8/24) Consider the road system shown below EADFORD TUAM CLAREGALWAY MOYCULLEN ATENRY SPIDDAL GALWAY ORANMORE Suppose there has been severe flooding, and Galway County Council can only keep a small number of roads open? Which ones should they choose, so that one can travel between any pair of towns?

9 Trees Spanning trees (9/24) Spanning tree Given a (simple) graph G, a SPANNING TREE of G is a subgraph of G that is a tree, and contains every vertex of G. EADFORD TUAM EADFORD TUAM CLAREGALWAY CLAREGALWAY MOYCULLEN MOYCULLEN ATENRY ATENRY SPIDDAL GALWAY SPIDDAL GALWAY ORANMORE ORANMORE Lots of other spanning trees are possible, and there are numerous ways of finding them...

10 Trees Spanning trees (10/24) Lots of other spanning trees are possible, and there are numerous ways of finding them. ere are two: Algorithm 1 (i) Identify a cycle in the graph (ii) Delete an edge in that cycle, taking care not to disconnect the graph. (iii) Keep going until all cycles have been removed. Algorithm 2 (i) Start with just the vertices of the graph (no edges). (ii) Add an edge from the original graph, as long as it does not form a cycle. (iii) Stop when the graph is connected.

11 Trees Spanning trees (11/24) Example (MA204, Semester 1 Exam, 2012) Describe an algorithm for finding a spanning tree in a connected graph and use the algorithm to find such a spanning tree in the connected graph illustrated below: f e d a c b

12 Trees Spanning trees (12/24) There are many other applications of tress that, regrettably, we do not have time to cover. The most important of these include minimum spanning trees. the study of search algorithms modelled as trees; decision tress (like the puzzle from Slide 7); compiler syntax.

13 What we didn t study (13/24) Other topics in combinatorics and graph theory that we have not cover. The most interesting (to my mind are) directed graphs the representation of graphs using adjacency and incidence matrices; Algorithms, like determining if a graph is connected, or finding the shortest path between two vertices... the graph Lapacian; visualisation of graphs; and many, many, more...

14 What we didn t study Directed graphs (14/24) Graphs often represent networks, such as the road network we had earlier, or social networks. So far, we have had that, if vertex a is adjacent to vertex b, then b is adjacent to a. In many situations, this is not reasonable: a city road system might have a one-way system; on a social network, you might follow someone who does not follow you back. f f e d e d a c a b b c

15 What we didn t study Directed graphs (15/24) Example: Graph of a tournament (2017 Senior Men s 6 Nations Rugby) FRANCE ITALY ENGLAND SCOTLAND WALES IRELAND

16 What we didn t study Adjacency Matrix (16/24) In a practical setting, a graph must be stored in some computer-readable format. One of the most comment is an adjacency matrix. If the graph has n vertices, labelled {1, 2,..., n}, then the adjacency matrix is an n n matrix, A, with entries { 1 vertex i is adjacent to j a i,j = 0 otherwise

17 What we didn t study Adjacency Matrix (17/24) Properties of the adjacency matrix The adjacency matrix of a graph is symmetric. The adjacency matrix of a directed graph is not necessarily symmetric. If B = A k, then b i,j is the number of paths from vertex i to vertex j. We can work out if a graph is connected by looking at the eigenvalues of A. If the graphs G and are isomorphic, and have adjacency matrices A G and A h, respectively, then there is a permutation matrix, P, such that PA G P 1 = A.

18 Review (18/24) But the set of topics that we did study includes: 1. The additive and multiplicative principles; 2. Sets, including power sets; union and intersection; 3. the Principle of Inclusion/Exclusion (PIE) and its applications; 4. Binomial Coefficients (& lattice paths, bit-strings); Pascal s triangle; 5. Permutations and Combinations; 6. Stars and Bars, multisets, and the NNI Equations and Inequalities; 7. Algebraic and Combinatorial Proofs; 8. Derangements; counting with repetition; 9. Graph Theory: motivation and basic definitions; 10. Isomorphisms between graphs. 11. Important families of graphs (Cycle graphs, K n, K n,n, etc.) 12. Planar & non-planar graphs; chromatic numbers, Euler s formula, Convex polyhedra, and Platonic solids; 14. Graph Colouring; Greedy and Welsh-Powell algorithms; 15. Eulerian and amiltonian graphs; 16. Trees.

19 Review The Final Exam (19/24) The final exam for Discrete Mathematics: 1 There are 10 questions: you should attempt all ten. 2 There are 5 questions on combinatorics, and 5 on graph theory Tips:

20 Review Past exam papers (20/24) Up until 2014/2015, Discrete Mathematics was delivered as two separate modules: MA284 and MA204. It s current incarnation is similar, but is not identical to either. Some of the past exam papers for them are useful study aids: 2016/2017 MA204/MA284: Everything 2015/2016 MA204/MA284: Everything except Q4(c). 2014/2015 MA284 Q1, Q2, Q3, Q4(b), but not Q4(a) [Platonic graphs] and Q4(c) [m-ary trees] 2014/2015 MA204 Q1; Q2(a), (b) and (d); Q3(a) and (b), Q4 but not Q2(c) [recurrence]; Q3(c) [Reverse Polish Notation]

21 Review Revision Questions (21/24) Q1. Find the number of different arrangements of the letters in the word: MISAPPREENSION. Of these arrangements, (a) how many have all the vowels together? (b) how many start and end with N? (c) how many start or end with N? (d) how many have all the letters in alphebetical order? Q2. (a) The sets A and B are such that A = 10 and B = 20. What is the largest possible value of A B? What is the largest possible value of A B? What is the value of A B + A B? (b) Suppose that four sets are such that each set has 30 elements; each pair of sets share 10 elements, and each triple of sets share 5 elements. If the union of all four sets has 80 elements, how many elements are there in the intersection of all four sets? Q3. (a) Prove that k ( n) ( k = n n 1 ( k 1). (b) Prove that n ) ( k = n 1 ) ( k 1 + n 1 ) k. Q4. (a) ow many (binary) bit strings are there of length 8? ow many of these have weight 3?

22 Review Revision Questions (22/24) (b) A ternary string is a sequence of 0s, 1s, and 2s. ow many ternary strings of length 17 are there? ow many of those strings contain exactly eight 0s, four 1s, and five 2s? ow many ternary strings of length 17 contain an odd number of 1s? Q5. ow many non-negative integer solutions are there to the equation x 1 + x 2 + x 3 + x 4 + x 5 < 11, if there are no restrictions? ow many solutions are there if x 1 > 3? ow many solutions are there if each x i < 3? Q6. Students in an Indiscreet Mathematics class work together in groups of 5 for an assignment. The group is given a score, which they divide up, according to the amount of work each did, to get their individual scores. Aoife, Brian, Conor, Declan, and Eimer worked together, and got a score of 20. (a) ow many ways can their scores be assigned? (b) The were given scores (respectively) of 2, 4, 6, 8 and 0. The lecturer entered these scores, but assigned them all to the wrong people. ow many ways can this happen? Q7. (a) Prove that v V deg(v) = 2 E for any graph G = (V, E). Deduce that the number of edges in the complete graph on n vertices is equal to ( n 2). (b) Prove that if a connected planar graph has v vertices, e edges, and f faces, then v e + f = 2. Use this to show that K 3,3 is not planar.

23 Review Revision Questions (23/24) Q8. Determine the chromatic number of each of the following graphs, and give a corresponding colouring. Q9. Explain the terms Eulerian path and Eulerian circuit. For each of the following graphs, determine if it has an Eulerian path and/or Eulerian circuit. If so, give an example; if not, explain why. (a) G = (V, E) with V = {a, b, c, d, e, f } and E = { {a, b}, {a, c}, {a, d}, {a, f }, {b, c}, {b, d}, {b, e}, {c, e}, {c, f }, {d, e}, {d, f }, {e, f } } (b) g c i j e d f a b h Q10. (a) Show that if T is a tree with e edges, then it has e + 1 vertices. (b) Show that if T is an acyclic graph with v vertices, and e = v 1 edges, then it is a tree.

24 A summary in one slide (1/2)... (24/24)