0.0.1 Network Analysis

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1 Graph Theory Network Analysis Prototype Example: In Algonquian Park the rangers have set up snowmobile trails with various stops along the way. The system of trails is our Network. The main entrance is on highway 11 at station O. The big chalet is at station T and

2 the only way in is via those trails. A-F are the stops that have little shelters and phones. The distances between stops (nodes) are shown. There are limits on each trail s (arc) maximum throughput. For example the edge OA can have a maximum of 10 users per hour (flow) to prevent trail damage and scaring the animals. The questions we can ask are: What s the shortest path from the highway to the chalet? (Shortest Path 2

3 Problem) What s the maximum number of sleds that can go from the highway to the chalet? (Maximum Flow Problem) Say I needed to wire the whole trail system with phone lines. What s the cheapest way to connect all nodes to each other? (Minimum Spanning Tree Problem) Each of these problems will be studied in turn. Each has its own algorithm. 3

4 Graph finite set of points, vertices, andaset of line segments, edges, to join pairs of vertices. Adjacent vertices are those joined by an edge. Nonadjacent are those that are not joined by a single edge. The degree, d(v)ofthe vertex is the number of edges emanating from it. What s the relation between number of edges and the 4

5 sum of the degrees of the vertices? What is the parity of the sum of degrees of the vertices? A cycle is a path that returns to a vertex without using an edge twice. Atreeisagraphwithno cycles 0.1 Minimum Spanning Tree Problem If a network has n nodes, a spanning tree has (n-1) arcs 5

6 and no cycles. A spanning tree allows a path from any node to any other node. Some examples of what s not spanning trees. The algorithm for this problemisasfollows: 1) pick any node, add a link to the nearest node. 2) pick the nearest unconnected node to one of your connected nodes, add that link. 3) break ties arbitrarily, stop when there are no unconnected nodes left. 6

7 Note that ties might mean multiple solutions. Example on board. Some more terms: Degree Sequence: a list of the degrees of a graphs vertices large to small. Multigraphs allows multiple edges between vertices, nuf said Pseudographs allows loops and multiple edges So all graphs are multigraphs and all multigraphs are 7

8 pseudographs Directed graphs (digraphs) have directed arcs, we can turn any graph into a digraph. Now we can talk about indegree and outdegree of a vertex. Of course the sum of the indegrees equals the sum of the out degrees Homework Section 8.1, # 1-8, Submit # 2a, 4d, 6c, 8d When the edges have no distances associated with them 8

9 we call the length of a path the number of edges in it. Thus apathwithk vertices has a length of k 1. Connected versus Disconnected graphs. Subgraphs a subgraph, A, of a larger graph, B, has a set of vertices V (A) V (B) and a set of edges E(A) E(B). GraphA is a subgraph of graph B is written as A B 9

10 Induced Subgraph an induced subgraph is a subgraph that is made by deleting vertices from the bigger graph. That is, all vertices in the subgraph are connected the same waytheyareinthebiggraph. Show some examples. A is a maximal subgraph of B with respect to property P if whenever C B having property P and E(A) E(C) E(B) and V (A) V (C) V (B) then A = C. 10

11 Example 1 Draw a disconnected graph. The pieces are called components. These components are maximal subgraphs with respect to the property connectedness. Example 2 Is a spanning tree a maximal subgraph with respect to the property of being a tree? To completely describe a graph we need the vertex set and the edge set. The degree sequence is not sufficient. Take for instance p-chloroflouro-benzene and n-chloro- 11

12 flouro-benzene. Isomorphic Graphs Two graphs are isomorphic if they are the same graphs with just re-labeled vertices. For example The relabeling function is called an isomorphism f : V (A) V (B). Go back to yesterday s example and define the isomorphism. No looking in your text! This is an example from the textbutitwasit saneatone. 12

13 Find the five non-isomorphic graphswiththedegreesequence 3,3,2,2,1,1 How to prove they are different graphs? The shortest path between two vertices and their mapped to vertices is the same length in the two isomorphic graph. How many possible molecules can made with 2 nitrogen, 2 oxygen and 2 hydrogen atoms? If two graphs are isomor- 13

14 phic then They have the same number of vertices They same number of components The same number of edges The same degree sequence The lengths of the shortest path between pairs of verticeswithagivendegreeare equal The lengths of the longest path are equal 14

15 0.2 Homework Section 8.2 #1, 2, 4, 5, 6a, Example 3 Draw the diagram for the graph G = G(V,E) where V = {a, b, c, d, e, f, g} and E = {ac,cd,fg,ga,bc,ad,cf,bc} Example 4 Formally describe the graph below. 15

16 1 Classes of Graphs We have already defined the term or class of graph called a tree. The following are equivalent: Gisatree G is connected and has n 1 edges with n vertices. Ghasn 1 edges and no cycles Any two vertices are connected by a unique path G has no cycles but the addition of one more edge 16

17 will produce a single cycle. Also remember that a disconnected graph with each component is a tree is called a forest. P n C n Other types path of n vertices cycle of n vertices Bipartite graphs Can be labelled with alternating symbols equivalent to a connected graph having no odd number cycles. 17

18 Draw a bipartite graph and one with an odd number cycle. Show why bipartite is equivalent to no odd number cycles. Complete Bipartite Graph Every vertex is connected to every other vertex of a bipartite graph. We use K m,n to symbolize a complete bipartite graph with m vertices of the first type and n of the second. K n Complete Graph 18

19 all vertices adjacent to all of the others. They have Ã! n 2 = n 1 X i=1 edges for n vertices. Why? Draw K 4,P 3,C 3,K 1,K 2,K 3,4,K 1,4 Regular graphs a graph is k-regular if all its vertices havethesamedegree,k. How many edges does a k-regular graph have? Note that we have shown that k and n can t both be odd. 19 i

20 Thus K n is (n - 1)-regular. Planar graph Can be drawn without crossing edges. Is K 4,P n,c n,k 2,3 planar? Homework Section 8.3 # 1, 2a,c, 3, 4, 5 (Minimum Spanning Tree), 9, 12 20

21 1.1 Adjacency Matrix Given the graph below we can write a matrix that describes the connections between the vertices. 21

22 a ij = Call ( the matrix A, 1 if there is an edge from v i to v j 0 if there is no edge from v i to v j The matrix A is called the adjacency matrix. We can also use an adjacency matrix to completely describe a graph. Example 5 Draw the graph with the adjacency matrix A =

23 Does the adjacency matrix have to be symmetrical? Notice that there has to be 0 s in the diagonal. If the adjacencymatrixisnotsymmetric it could be a digraph, or it has 1 s in the diagonal it could be a pseudograph. Example 6 Draw the pseudodigraph 23

24 with the adjacency matrix A = These adjacency matrices are perfect for storing graphs in computers. They also let us investigate walks of any length. Consider the graph given by the 24

25 adjacency matrix 0101 A = Starting from the first vertex, where can you get to in a walk of two steps?, three steps? Calculate A 2,A 3 and infer that the entries of matrix A k tell the number of walks of length k between two vertices. We can even use A 2 to calculate the degree of all the vertices. How? 25

26 What do the diagonal elements of A 3 count? Thus 1 6 nx i=1 h A 3i ii = Use that fact to ascertain if the graph A =

27 is bipartite. Note that = A graph is bipartite if and only if it can be relabeled so that it can be written as Why? A = " 0 C C t 0 Draw K 3,4 and show it can be written in the matrix form above. 27 #

28 1.2 Incidence Matrix The incidence matrix, B, of a graph G(V,E) involves the edges, E = {e 1,e 2,...e q }, more that the vertices, V = {v 1,v 2 (,...v n } b ij = 1 if v i is an endpoint of e j 0 otherwise This matrix doesn t have to be square. What are the dimensions? Make the incidence matrix for 28

29 Notice that there are two 1 s in each column. What does the sum of the ith row represent? Sure B isn t square but B B t is square and of dimension n n, thesameasa. Whatis the connection between A and B? First what are the diagonals 29

30 of B B t? We can see that h B B ti ij = ( a ij i 6= j d(v i ) i = j Homework Section 8.4 #1,2,9Submit2andalsofind how many triangles are there in the graph shown in graph (a) on page

31 2 Traversing Graphs The Konigsberg Bridge Problem that you worked on for homework is an example of a traversing graph problem. You discovered that you couldn t draw a trail that crossed every bridge. Let s define a trail on agraphasawalkwherethe edges are distinct, that is you only use an edge once. Now we have walks, paths and trails A Path in a graph is a sequence of distinct vertices 31

32 AWalkinagraphisasequence of vertices A Trail in a graph is a walk where the edges are distinct. A Eulerian Trail is a trail that includes every edge of the graph, that is it uses every edge once and only once A Eulerian Circuit is a Eulerian trail that begins and ends at the same vertex A Path that contains every vertex is called a Hamiltonian path. 32

33 A Hamiltonian cycle is a cycle that contains every vertex. A graph that contains a closed Eulerian Trail or Eulerian Circuit is called a Eulerian Graph. See Boat example. A graph that contains a Eulerian trail is called semi- Eulerian. See House example. A graph that contains a Hamiltonian Cycle then it is a Hamiltonian graph. See House 33

34 A graph is called semi- Hamiltonian if it contains a Hamiltonian path. See K 2,3 A connected graph is Eulerian if and only if all the vertices have even degree. A connected graph is semi- Eulerian but not Eulerian iff it contains two vertices with odd degree and the rest have even degree. The Eulerian path starts at one odd vertex and ends at another. Thus Bridges can t be done. 34

35 Homework Section 8.6 #1,2,5, 3 Dijksrra s Algorithm (Not in Text) This is an algorithm that finds the length of the shortest path between vertices. First make an n n table and star the first settled node with a distance of zero. The first column will contain the distances from that first settled node to any other. If you can t 35

36 1. 36

37 get there in one step from a settled node then just put NA. Next settle the closest node. Complete the next column, and the next, until the chart is full. 37

38 Below is misc crap, my notes ignore Shortest Path Problem This method solves undirected connected networks. The objective is to find the shortest path from O, the supply node, to T, the demand node. Solved nodes are nodes that I have been to, unsolved nodes are nodes I haven t been to. List: Solved Nodes directly connected to unsolved nodes Closest Unsolved Node 38

39 Total Distance nearest node Minimum distance Last connection The Maximum Flow Problem Augmented Path Algorithm First draw the residual network with the capacities in and out at each node. A cut is a line drawn across the network separating the supply and demand node. The maximum flow through the cut is the minimum amount that 39

40 can be further pumped through the network. Try this on the example network. 1) pick a path through the network from O to T that uses all positive residuals.(augmented path) 2) choose the smallest residual through that chosen path. Pump that amount through that path, adjusting the residuals accordingly. 3) stop when there are no more augmented paths or you can find a minimum cut of 0. 40

41 This solution is then optimal. Know: max-flow min-cut theorem Do the example: You need to build a house as quickly as possible. The budget is $120,000. The times for each job are given below at the various speeds they can be done. Time Foundation Framing plumbing electrical normal priority crash

42 The costs for each state of urgency are given below. Money Foundation Framing plumbing electrical normal priority crash What s the fastest way to build the house? Formulate as a shortest path problem. 42