Download presentation

Presentation is loading. Please wait.

Published byDouglas Emil Lyons Modified over 6 years ago

2
Graph Graph Types of Graphs Types of Graphs Data Structures to Store Graphs Data Structures to Store Graphs Graph Definitions Graph Definitions

3
A graph is a pictorial representation of data. A graph connects different vertices and provides a flow of data. Back

4
Definition (undirected, unweighted): ◦ A Graph G, consists of ◦ a set of vertices, V ◦ a set of edges, E ◦ where each edge is associated with a pair of vertices. We write: G = (V, E)

5
Directed Graph: ◦ Same as above, but where each edge is associated with an ordered pair of vertices.

6
Weighted Graph: ◦ Same as above, but where each edge also has an associated real number with it, known as the edge weight. Back

7
Adjacency Matrix Structure ◦ Certain operations are slow using just an adjacency list because one does not have quick access to incident edges of a vertex. ◦ We can add to the Adjacency List structure: a list of each edge that is incident to a vertex stored at that vertex. This gives the direct access to incident edges that speeds up many algorithms.

8
Adjacency Matrix ◦ The standard adjacency matrix stores a matrix as a 2-D array with each slot in A[i][j] being a 1 if there is an edge from vertex i to vertex j, or storing a 0 otherwise. 123456 1010010 2101010 3010100 4001011 5110100 6000100 Back

9
A complete undirected unweighted graph ◦ is one where there is an edge connecting all possible pairs of vertices in a graph. The complete graph with n vertices is denoted as K n. A graph is bipartite ◦ if there exists a way to partition the set of vertices V, in the graph into two sets V 1 and V 2 ◦ where V 1 V 2 = V and V 1 V 2 = , such that each edge in E contains one vertex from V 1 and the other vertex from V 2.

10
Complete bipartite graph ◦ A complete bipartite graph on m and n vertices is denoted by K m,n and consists of m+n vertices, with each of the first m vertices is connected to all of the other n vertices, and no other vertices.

11
A weighted graph ◦ A weighted graph associates a label (weight) with every edge in the graph. The weight of a path or the weight of a tree in a weighted graph is the sum of the weights of the selected edges. The function dist(v,w) ◦ The function dist(v, w), where v and w are two vertices in a graph, is defined as the length of the shortest weight path from v to w. dist(b,e) = 8

12
A subgraph ◦ A graph G'= (V', E') is a subgraph of G = (V, E) if V' V, E' E, and for every edge e' E', if e' is incident on v' and w', then both of these vertices are contained in V'.

13
A simple path ◦ A simple path is one that contains no repeated vertices. A cycle ◦ A path of non-zero length from and to the same vertex with no repeated edges. A simple cycle ◦ A cycle with no repeated vertices except for the first and last ones. 6 4 5 1 2 3

14
A path ◦ A path of length n from vertex v 0 to vertex v n is an alternating sequence of n+1 vertices and n edges beginning with vertex v 0 and ending with vertex v n in which edge e i incident upon vertices v i-1 and v i. (The order in which these are connected matters for a path in a directed graph in the natural way.) A connected graph ◦ A connected graph is one where any pair of vertices in the graph is connected by at least one path Back

15
THANK YOU Back

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google