Downloaded 1,244 times
![2. Incidence Matrix
VxE
[vertex, edges] contains the edge's data
1, 2
1,5
2 ,3
2 ,5
3, 4
4 ,5
4 ,6
1
1
1
0
0
0
0
0
2
1
0
1
1
0
0
0
3
0
0
1
0
1
0
0
4
0
0
0
0
1
1
1
5
0
1
0
1
0
1
0
6
0
0
0
0
0
0
1](https://image.slidesharecdn.com/1introductiontographtheory-140220082526-phpapp02/75/Introduction-to-Graph-Theory-22-2048.jpg)
Uploaded byPremsankar Chakkingal
Introduction to Graph Theory
The document provides an introduction to graph theory. It begins with a brief history, noting that graph theory originated from Euler's work on the Konigsberg bridges problem in 1735. It then discusses basic concepts such as graphs being collections of nodes and edges, different types of graphs like directed and undirected graphs. Key graph theory terms are defined such as vertices, edges, degree, walks, paths, cycles and shortest paths. Different graph representations like the adjacency matrix, incidence matrix and adjacency lists are also introduced. The document is intended to provide an overview of fundamental graph theory concepts.
More Related Content
![Depth first search [dfs]](https://cdn.slidesharecdn.com/ss_thumbnails/depthfirstsearchdfs-190926145304-thumbnail.jpg?width=640&height=640&fit=bounds)
![EE-304 Electrical Network Theory [Class Notes1] - 2013](https://cdn.slidesharecdn.com/ss_thumbnails/ee30420130802graph-130917043018-phpapp01-thumbnail.jpg?width=640&height=640&fit=bounds)
Similar to Introduction to Graph Theory
Introduction to Graph Theory
- 1. Introduction to GraphTheory HANDBOOK OF GRAPH THEORY FOR FRESHER'S Prem Sankar C M Tech Technology Management Dept of Futures Studies ,Kerala University
- 2. Outline 1. 2. 3. 4. 5. History of GraphTheory Basic Concepts of Graph Theory Graph Representations Graph Terminologies Different Type of Graphs
- 3. Why Graph Theory? Graphs used to model pair wise relations between objects Generally a network can be represented by a graph Many practical problems can be easily represented in terms of graph theory
- 4. Graph Theory -History The origin of graph theory can be traced back to Euler's work on the Konigsberg bridges problem (1735), which led to the concept of an Eulerian graph. The study of cycles on polyhedra by the Thomas P. Kirkman (1806 - 95) and William R. Hamilton (1805-65) led to the concept of a Hamiltonian graph.
- 5. Graph Theory -History Begun in 1735 Mentioned in Leonhard Euler's paper on “Seven Bridges of Konigsberg ” . Problem : Walk all 7 bridges without crossing a bridge twice
- 6. Graph Theory –History……. Cycles in Polyhedra - polyhedron with no Hamiltonian cycle Thomas P. Kirkman William R. Hamilton Hamiltonian cycles in Platonic graphs
- 7. Graph Theory –History….. Trees in Electric Circuits Gustav Kirchhoff
- 8. Basic Concepts ofGraph Theory
- 9. Definition: Graph Agraph is a collection of nodes and edges Denoted by G = (V, E). V = nodes (vertices, points). E = edges (links, arcs) between pairs of nodes. Graph size parameters: n = |V|, m = |E|.
- 10. Vertex & Edge Vertex /Node Basic Element Drawn as a node or a dot. Vertex set of G is usually denoted by V(G), or V or VG Edge /Arcs A set of two elements Drawn as a line connecting two vertices, called end vertices, or endpoints. The edge set of G is usually denoted by E(G), or E or EG Neighborhood For any node v, the set of nodes it is connected to via an edge is called its neighborhood and is represented as N(v)
- 11. Graph :Example n:=6 , m:=7 Vertices (V) :={1,2,3,4,5,6} {1,2},{1,5},{2,3},{2,5},{3,4},{4,5},{4,6}} N(4) := Neighborhood (4) ={6,5,3} Edge (E) :=
- 12. Edge types: Undirected; Directed; orderedpairs of nodes. E.g., distance between two cities, friendships… E.g ,… Directed edges have a source (head, origin) and target (tail, destination) vertices Weighted ; usually weight is associated .
- 13. Empty Graph /Edgeless graph No edge Null graph No nodes Obviously no edge
- 14. Simple Graph (Undirected) Simple Graph are undirected graphs without loop or multiple edges A = AT F or sim ple graphs, deg( v i ) vi V 2|E |
- 15. Directed graph :(digraph) Edges have directions A !=AT loop multiple arc arc node
- 16. Weighted graph isa graph for which each edge has an associated weight 1 2 1.2 2 3 .2 .3 .5 4 1.5 5 .5 1 6 5 1 4 3 2 5 3 6
- 17. Bipartite Graph V canbe 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.
- 18. Trees An undirectedgraph is a tree if it is connected and does not contain a cycle (Connected Acyclic Graph) Two nodes have exactly one path between them
- 19. Subgraph Vertex andedge sets are subsets of those of G a supergraph of a graph G is a graph that contains G as a subgraph.
- 20.
- 21. 1. Adjacency Matrix n-by-n matrix with Auv = 1 if (u, v) is an edge. Diagonal Entries are self-links or loops Symmetric matrix for undirected graphs 1 2 3 4 5 6 7 8 1 0 1 1 0 0 0 0 0 2 1 0 1 1 1 0 0 0 3 1 1 0 0 1 0 1 1 4 0 1 0 1 1 0 0 0 5 0 1 1 1 0 1 0 0 6 0 0 0 0 1 0 0 0 7 0 0 1 0 0 0 0 1 8 0 0 1 0 0 0 1 0
- 22. 2. Incidence Matrix VxE [vertex,edges] contains the edge's data 1, 2 1,5 2 ,3 2 ,5 3, 4 4 ,5 4 ,6 1 1 1 0 0 0 0 0 2 1 0 1 1 0 0 0 3 0 0 1 0 1 0 0 4 0 0 0 0 1 1 1 5 0 1 0 1 0 1 0 6 0 0 0 0 0 0 1
- 23. 3. Adjacency List Edge List Edge List 12 12 23 25 33 43 45 53 54 Adjacency List (node list) Node List 122 235 33 435 534
- 24. Edge Lists forWeighted Graphs Edge List 1 2 1.2 2 4 0.2 4 5 0.3 4 1 0.5 5 4 0.5 6 3 1.5
- 25.
- 26. Classification of GraphTerms Global terms refer to a whole graph Local terms refer to a single node in a graph
- 27. Connected and Isolatedvertex Two vertices are connected if there is a path between them Isolated vertex – not connected 1 isolated vertex 2 3 4 5 6
- 28. Adjacent nodes Adjacentnodes -Two nodes are adjacent if they are connected via an edge. If edge e={u,v} ∈ E(G), we say that u and v are adjacent or neigbors An edge where the two end vertices are the same is called a loop, or a self-loop
- 29. Degree (Un DirectedGraphs) Number of edges incident on a node The degree of 5 is 3
- 30. Degree (Directed Graphs) In-degree:Number of edges entering Out-degree: Number of edges leaving Degree = indeg + outdeg outdeg(1)=2 indeg(1)=0 outdeg(2)=2 indeg(2)=2 outdeg(3)=1 indeg(3)=4
- 31. Walk trail: noedge can be repeat a-b-c-d-e-b-d walk: a path in which edges/nodes can be repeated. a-b-d-a-b-c A walk is closed is a=c
- 32. Paths Path: isa sequence P of nodes v1, v2, …, vk-1, vk No vertex can be repeated A closed path is called a cycle The length of a path or cycle is the number of edges visited in the path or cycle 1,2,5,2,3,4 walk of length 5 Walks and Paths 1,2,5,2,3,2,1 CW of length 6 1,2,3,4,6 path of length 4
- 33. Cycle Cycle -closed path: cycle (a-b-c-d-a) , closed if x=y Cycles denoted by Ck, where k is the number of nodes in the cycle C3 C4 C5
- 34. Shortest Path ShortestPath is the path between two nodes that has the shortest length Length – number of edges. Distance between u and v is the length of a shortest path between them The diameter of a graph is the length of the longest shortest path between any pairs of nodes in the graph
- 35. THANK YOU Prem SankarC M Tech Technology Management Dept of Futures Studies Kerala University Prem Sankar C - Dept of Futures Studies