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Graphs

Graphs. Topics to be discussed…. What is a graph? Directed vs. undirected graphs Trees vs graphs Terminology: Degree of a Vertex Graph terminology Graph Traversal Graph representation. What is a graph?.

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Graphs

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  1. Graphs

  2. Topics to be discussed… • What is a graph? • Directed vs. undirected graphs • Trees vs graphs • Terminology: Degree of a Vertex • Graph terminology • Graph Traversal • Graph representation

  3. What is a graph? • A data structure that consists of a set of nodes (vertices) and a set of edges that relate the nodes to each other • The set of edges describes relationships among the vertices

  4. Formal definition of graphs • A graph G is defined as follows: G=(V,E) V(G): a finite, nonempty set of vertices E(G): a set of edges (pairs of vertices) back

  5. Directed vs. undirected graphs • When the edges in a graph have no direction, the graph is called undirected

  6. Directed vs. undirected graphs (cont.) • When the edges in a graph have a direction, the graph is called directed (or digraph) Warning: if the graph is directed, the order of the vertices in each edge is important !! E(Graph2) = {(1,3) (3,1) (5,9) (9,11) (5,7) back

  7. Trees vs graphs • Trees are special cases of graphs!! back

  8. Terminology:Degree of a Vertex • The degree of a vertex is the number of edges incident to that vertex • For directed graph, • the in-degree of a vertex v is the number of edgesthat have v as the head • the out-degree of a vertex v is the number of edgesthat have v as the tail • if di is the degree of a vertex i in a graph G with n vertices and e edges, the number of edges is

  9. Examples 0 3 2 1 2 0 3 3 1 2 3 3 6 5 4 3 1 G1 1 1 3 G2 1 3 0 in:1, out: 1 directed graph in-degree out-degree in: 1, out: 2 1 in: 1, out: 0 2 G3 back

  10. Graph terminology • Adjacent nodes: two nodes are adjacent if they are connected by an edge • Path: a sequence of vertices that connect two nodes in a graph • Complete graph: a graph in which every vertex is directly connected to every other vertex

  11. Graph terminology (cont.) • What is the number of edges in a complete undirected graph with N vertices?  N * (N-1) / 2

  12. Graph terminology (cont.) • Weighted graph: a graph in which each edge carries a value back

  13. Graph Traversal • Problem: Search for a certain node or traverse all nodes in the graph • Depth First Search • Once a possible path is found, continue the search until the end of the path • Breadth First Search • Start several paths at a time, and advance in each one step at a time back

  14. Graph Representations • Adjacency Matrix • Adjacency Lists

  15. Adjacency Matrix • Let G=(V,E) be a graph with n vertices. • The adjacency matrix of G is a two-dimensional n by n array, say adj_mat • If the edge (vi, vj) is in E(G), adj_mat[i][j]=1 • If there is no such edge in E(G), adj_mat[i][j]=0 • The adjacency matrix for an undirected graph is symmetric; the adjacency matrix for a digraph need not be symmetric

  16. 0 4 1 2 5 6 3 7 0 0 1 2 3 1 2 G2 G1 symmetric undirected: n2/2 directed: n2 G4 back

  17. Adjacency Lists (data structures) Each row in adjacency matrix is represented as an adjacency list.

  18. 0 0 4 1 2 5 3 6 7 1 2 3 0 1 2 3 1 1 2 3 2 0 1 2 3 4 5 6 7 0 2 3 0 3 0 1 3 0 3 1 2 1 2 0 5 G1 0 6 4 0 1 2 1 5 7 1 0 2 6 G4 G3 2 back An undirected graph with n vertices and e edges ==> n head nodes and 2e list nodes

  19. Thank You

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