Lecture 16 graph introduction
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1) A graph is a collection of vertices and edges, where vertices represent nodes and edges connect pairs of vertices. 2) Graphs can be undirected, where edges have no directionality, or directed (digraphs), where edges have a direction associated with them. 3) Graphs are commonly represented using adjacency matrices, with cells indicating connections between vertex pairs, or adjacency lists, with each vertex listing its connected neighbors.
Lecture 16 graph introduction
- 1. Lecture 16 Introduction to Graph Abirami Sivaprasad
- 2. Vertices and Edges Definition: A graph is a collection (nonempty set) of vertices and edges Vertices: can have names and properties(set of nodes ) Edges: connect two vertices, can be labeled, can be directed Adjacent vertices: there is an edge between them
- 3. ExampleGraph1 Vertices: A,B,C,D Edges: AB, AC, BC, CD A B C D A C B D Two ways to draw the same graph
- 4. Undirected graphs When the edges in a graph have no direction, the graph is called undirected
- 5. Directed Graph When the edges in a graph have a direction, the graph is called directed (or digraph) Note: if the graph is directed, the order of the vertices in each edge is important !!
- 6. More definitions : Path A list of vertices in which successive vertices are connected by edges A B C B A C D A B C A B C A B C D B A B A C A B C D
- 7. More definitions : Simple Path No vertex is repeated. A B C D D C A D C B A B A B C A B C D
- 8. More definitions : Cycle Simple path with distinct edges, except that the first vertex is equal to the last A B C A B A C B C B A C A B C D A graph without cycles is called acyclic graph.
- 9. More definitions : Loop An edge that connects the vertex with itself A B C D
- 10. Connected and Disconnected graphs Connected graph: There is a path between each two vertices Disconnected graph : There are at least two vertices not connected by a path. Examples of disconnected graphs: A B C D A B C D
- 11. Graphs and Trees Tree: an undirected graph with no cycles, and a node chosen to be the root A B C E D Source graph:
- 12. Graphs and Trees A C E B D C A E B D Tree1: root A Tree2: root C
- 13. A spanning tree of an undirected graph A sub-graph that contains all the vertices, and no cycles. If we add any edge to the spanning tree, it forms a cycle, and the tree becomes a graph A B C D A B C D graph spanning tree
- 14. Examples A B C D A B C D A B C D All spanning trees of the graph on the previous slide
- 15. Complete graphs Graphs with all edges present – each vertex is connected to all other vertices Dense graphs: relatively few of the possible edges are missing Sparse graphs: relatively few of the possible edges are present A B C D E A complete graph
- 16. Weighted graphs and Networks Weighted graphs – weights are assigned to each edge (e.g. road map) Networks: directed weighted graphs (some theories allow networks to be undirected) A B D C 3 2 4 1 2
- 18. Adjacency matrix – undirected graphs Vertices: A,B,C,D Edges: AC, AB, AD, BD A B C D A 0 1 1 1 B 1 0 0 1 C 1 0 0 1 D 1 1 0 0 The matrix is symmetrical A B C D
- 19. Adjacency matrix – directed graphs Vertices: A,B,C,D Edges: AC, AB, BD, DA A B C D A 0 1 1 0 B 0 0 0 1 C 0 0 0 0 D 1 0 0 0 A B C D
- 20. Adjacency lists – undirected graphs Vertices: A,B,C,D Edges: AC, AB, AD, BD Heads lists A B C D B A D C A D A B A B C D
- 21. Adjacency lists – directed graphs Vertices: A,B,C,D Edges: AC, AB, BD, DA Heads lists A B C B D C = D A A B C D
- 22. Thank u