Graphs Topics to be discussed What is a

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? 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

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

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

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(Graph 2) = {(1, 3) (3, 1) (5, 9) (9, 11) (5, 7) back

Trees vs graphs Trees are special cases of graphs!! back

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 edges that have v as the head the out-degree of a vertex v is the number of edges that 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

Examples 0 3 2 1 0 3 1 3 2 3 3 directed graph in-degree out-degree 3 5 6 1 0 1 G 2 1 in: 1, out: 1 1 in: 1, out: 2 2 in: 1, out: 0 4 1 3 G 1 3 2 G 3 back

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

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

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

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

Graph Representations Adjacency Matrix Adjacency Lists

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

0 1 0 2 4 0 3 1 5 1 2 6 3 7 2 G 1 symmetric undirected: n 2/2 directed: n 2 G 4 back

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

0 4 0 1 1 2 2 6 3 3 0 1 2 3 1 0 0 0 7 2 2 1 1 G 1 0 1 2 1 0 2 G 3 5 3 3 3 2 0 1 2 3 4 5 6 7 2 An undirected graph with n vertices and e edges ==> n head nodes and 2 e list nodes 1 0 0 1 5 4 5 6 G 4 2 3 3 2 6 7 back

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