1 / 19

Understanding Graph Connectivity and Terminology in Computer Science

This lecture covers the fundamentals of graph theory, including terminology, types of graphs, and vertex degrees. Key concepts such as adjacency, incidence, cycles, and graph representations are introduced. Students will learn about various graph types such as simple graphs, multigraphs, directed graphs, and bipartite graphs, along with examples. The importance of degrees of vertices, including in-degree and out-degree, is explained, as well as concepts like isomorphism and necessary conditions for graph similarity. Assignments and projects further reinforce these concepts.

marla
Télécharger la présentation

Understanding Graph Connectivity and Terminology in Computer Science

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 19: CONNECTIVITYSections 8.1 - 8.3 CS1050: Understanding and Constructing Proofs Spring 2006 Jarek Rossignac

  2. Lecture Objectives • Learn graph terminology

  3. What are the types of graph? Graph G(V,E) • V = set of vertices (non-empty) • E = set of edges (unordered pairs of distinct elements of V) Loop Multiple edge Simple Graph Multigraph Pseudograph Directed graph

  4. Graph Types

  5. Examples of graphs • Simple: • Multigraph (multiple edges): • Pseudograph (multiple edges and loops): • Directed (loops): • Directed multigraph (multiple edges and loops):

  6. Classify graphs Edge between A and B means: • They know each other • A is a parent of B • They compete • A has called B • Page A has a link to page B • Have collaborated • A has beaten B in round-robin

  7. What is adjacency and incidence? In an undirected graph An edge E between vertices A and B is incident with them. A and B are the endpoints of E E connects A and B Vertices A and B are adjacent (neighbors) when there is an edge incident with both

  8. What is the degree of a vertex? In an undirected graph with e edges: The degree deg(V), also called valence, of vertex V is the number of times V is used by an edge (twice by an incident loop). A vertex with degree one is pendent (dead end). A vertex with degree zero is isolated. The sum of the degrees of all vertices if 2e. There is an even number of edges of odd degree.

  9. Directed graph terminology E is a directed edge from A to B (denoted AB) • A is adjacent to B • A is the initial vertex of E • B is adjacent from A • B is the terminal or end vertex of E A=B if E is a loop In-degree deg–(V) of vertex V is the number of edges for which it is a terminal vertex Out-degree deg+(V) of vertex V is the number of edges for which it is an initial vertex

  10. Cycles A cycle Cn is has n vertices and n-edge the form a cycle C3 is a triangle C5

  11. Complete graphs Kn A complete graph Kn of n vertices is a simple graph with one edge between each pair K3 is a triangle K5

  12. Wheels A wheel Wn is a cycle with n vertices plus an additional vertex connected to all W5

  13. Bipartite graphs A graph is bipartite when itd vertices can be colored (red/green) so that each edge joins vertices of different colors It is complete bipartite if there is an edge between each pair of vertices of different color

  14. Subgraph A subgraph of G has a subset of the edges and vertices of G • It must include all the vertices bounding all its edges!

  15. Representing graphs • Vertices (x,y) , edges (a,b) • Adjacency list: vertices (x, y, a, b, …) • Adjacency matrix • Simple graphs (binary, symmetric) • Multiple graph: integer entries count number of edges • Loops on diagonal • Incidence matrix: edges/vertices • Two 1s per column

  16. Isomorphism Two graphs G and H are isomorphic if there is a bijection between their vertices that leads to the same set of edges. Expensive to compute, since there are n! vertex/label assignments Necessary conditions (invariants) help quickly decide that two graphs are NOT isomorphic • same number of vertices and edges • same degree list

  17. Assigned Reading • 8.1, 8.2, 8.3

  18. Assigned Homework • P 544-545: 3, 4, 5, 6, 7 • P 555: 12, 27, 29f, 36, 42 • P 562: 1, 10, 38, 39, 49, 57a, 68

  19. Assigned Project • P9: Spanning tree

More Related