1. Graph Theory Chapter 6 from Johnsonbaugh Article(6.1, 6.2)

2. History Konisgsberg problem Wyoming Inspectores problem

3. Konigsberg bridge problem

4. Solution to Konisgbers problem Presented by Euler Representation of problem as a graph Presence of Euler Cycle…??

5. Some applications of Graph Theory Models for communications and electrical networks Models for computer architectures(Manufacturing) Network optimization models for operations analysis, including scheduling and job assignment Analysis of Finite State Machines Parsing and code optimization in compilers

6. Application to Ad Hoc Networking Networks can be represented by graphs The mobile nodes are vertices The communication links are edges Vertices Edges Routing protocols often use shortest path algorithms

7. Elementary Concepts A graph G(V,E) is two sets of object  Vertices (or nodes), set V  Edges, set E A graph is represented with dots or circles (vertices) joined by lines (edges) The magnitude of graph G is characterized by number of vertices |V| (called the order of G) and number of edges |E| (size of G) The running time of algorithms are measured in terms of the order and size

8. Graphs: Examples a c d b Let V={a,b,c,d} and E={{a,b},{a,c},{b,c},{c,d}}

9. Graphs ↔ Networks Graph (Network) Vertexes (Nodes) Edges (Arcs) Flow Communications Telephones exchanges, computers, satellites Cables, fiber optics, microwave relays Voice, video, packets Circuits Gates, registers, processors WiresCurrent Mechanical JointsRods, beams, springsHeat, energy Hydraulic Reservoirs, pumping stations, lakes PipelinesFluid, oil Financial Stocks, currencyTransactionsMoney Transportation Airports, rail yards, street intersections Highways, railbeds, airway routes Freight, vehicles, passengers

10. Undirected Graph 2 4 3 1 V = { 1, 2, 3, 4}, | V | = 4 E = {(1,2), (2,3), (2,4), (4,1)}, | E |=4 An edge e  E of an undirected graph is represented as an unordered pair (u,v)=(v,u), where u, v  V. Also assume that u ≠ v

11. Directed Graph An edge e  E of a directed graph is represented as an ordered pair (u,v), where u, v  V. Here u is the initial vertex and v is the terminal vertex. Also assume here that u ≠ v 2 4 3 1 V = { 1, 2, 3, 4}, | V | = 4 E = {(1,2), (2,3), (2,4), (4,1), (4,2)}, | E |=5

12. Definitions… Parallel edge Loop Isolated vertex Simple graph

13. Degree of a Vertex Degree of a vertex in an undirected graph is the number of edges incident on it. In a directed graph, the out degree of a vertex is the number of edges leaving it and the in degree is the number of edges entering it 2 4 3 1 2 4 3 1 The degree of vertex 2 is 3 The in degree of vertex 2 is 2 and the in degree of vertex 4 is 1

14. Weighted Graph A weighted graph is a graph for which each edge has an associated weight, usually given by a weight function w: E  R 2 4 3 1 2 4 3 1 1.2 2.1 0.2 0.5 4 8 6 2 9

15. Walks and Paths 3 2 3 4 1 1 V5V5 V4V4 V3V3 V2V2 V1V1 V6V6 4 1 A path is a sequences of edges that begins at a vertex of a graph and travels along edges of the graph always connecting pairs of adjacent vertices. A simple cycle is an walk (v 1, v 2,..., v L ) where v 1 =v L with no other nodes repeated and L>=3, e.g. (V 1, V 2,V 5, V 4,V 1 ) A simple path is a walk with no repeated nodes, e.g. (V 1, V 4,V 5, V 2,V 3 ) A graph is called cyclic if it contains a cycle; otherwise it is called acyclic

16. Definitions.. Cycle(circuit) Simple cycle Length of a path

17. Similarity Graph Example 6.1.7 The n-cube hyperbola

18. Complete Graphs A D C B 4 nodes and (4*3)/2 edges V nodes and V*(V-1)/2 edges C A B 3 nodes and 3*2 edges V nodes and V*(V-1) edges A complete graph is an undirected/directed graph in which every pair of vertices is adjacent. If (u, v ) is an edge in a graph G, we say that vertex v is adjacent to vertex u.

19. Connected Graphs A D E F BC A B C D An undirected graph is connected if you can get from any node to any other by following a sequence of edges OR any two nodes are connected by a path A directed graph is strongly connected if there is a directed path from any node to any other node

20. Subgraph components

21. Bipartite Graph A bipartite graph is an undirected graph G = (V,E) in which V can be partitioned into 2 sets V1 and V2 such that ( u,v)  E implies either u  V1 and v  V2 OR v  V1 and u  V2. u1u1 u2u2 u3u3 u4u4 v1v1 v2v2 v3v3 V1V1 V2V2 An example of bipartite graph application to telecommunication problems can be found in, C.A. Pomalaza-Ráez, “A Note on Efficient SS/TDMA Assignment Algorithms,” IEEE Transactions on Communications, September 1988, pp. 1078-1082. For another example of bipartite graph applications see the slides in the Addendum section

22. Euler cycle and Konigsberg problem Returning to Konigsberg Bridge problem There will be no euler cycle if there are an odd number of edges incident on vertex A Theorem:”If a graph has an Euler cycle, then G is connected and every vertex has even degree. Theorem:” If G is a connected graph and every vertex has even degree, then G has an Euler cycle

23. Theorem:” If G is a graph with m edges and vertices{v1, v2,……, v3} then In particular, the sum of the degrees of all the vertices in a graph is even.

24. Euler Cycle How to find an Euler Cycle….. Euler Path: “An Euler Path in G is a simple path containing every edge in G”

25. Some theorems and corollaries about graphs Corollary: “In any graph, there are an even number of vertices of odd degree” Theorem:” A graph has a path with no repeated edges from v to w( v is not equal to w) containing all the edges and vertices if and only if it is connected and v and w are the only vertices having odd degree. Theorem:” If graph G contains a cycle from v to v, G contains a simple cycle from v to v.

26. Announcements!!!! Assignment: Calculate the running time for Bubble Sort Algorithm Home Work: (Exercise 6.1 and 6.2 relevent questions) Quiz(in Next Lecture from todays lecture topics)