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What is a graph

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What is a graph

1. 1. What is a Graph? A graph is a nonlinear data structure. A graph data structure consists of a finite set of ordered pairs, called edges or arcs, of certain entities called nodes or vertices. The size of a graph is the number of elements in its vertex set. Figure shows a simple graph (V, E) of size 4. Its vertex set is V = {a, b, c, d}, and its edge set is E = {ab, ac, ad, bd, cd}. This graph has four vertices and five edges. Types of Graph: There are two types of graphs: i. Undirected Graphs ii. Directed Graphs Undirected Graph: A graph that entail edges with ordered pair of vertices, however it does not have direction define. Directed Graph: A graph that entail edges with ordered pair of vertices and has direction indicated with an arrow. Degree _ In an undirected graph, the degree of a vertex u is the number of edges connected to u. E.g; in the above figure the degree of vertex 5 is two. _ In a directed graph, the out-degree of a vertex u is the number edges leaving u, and its in- degree is the number of edges ending at u. E.g; in the above figure the in-degree of vertex 5 is two and the out-degree is null.
2. 2. Complete Graph Figure shows the complete graph on the set V = {a, b, c, d}. Its edge set E is E = {ab, ac, ad, bc, bd, cd}. The number of edges in the complete graph on n vertices is n(n–1)/2. Subgraph Vertex and edge sets are subsets of those of G A supergraph of a graph G is a graph that contains G as a subgraph. Null graph A graph G=(V,E) where E=0 is said to be Null or Empty graph Trivial Graph: A graph with One vertex and no edge is called as a trivial graph. MultiGraph(Without Self Edge): The term multigraph refers to a graph in which multiple edges between vertices are permitted. A multigraph G = (V, E) asa sa V1
3. 3. is a graph which has the set of vetrices and multiple edges between vertices. MultiGraph(With Self Edge): A multidigraph is a directed graph which is permitted to have multiple edges, i.e., edges with the own,source and target vertices. Isomorphic graphs • Isomorphism – Two graphs are isomorphic, if they are structurally identical, Which means that they correspond in all structural details. – Formal vertex-to-vertex and edge –to-edge correspondence is called isomorphism. • Two graph are said to be isomorphic if – They have the same no of vertices. – They have the same number of edges. – They have an equal number of vertices with a given degree.
4. 4. Graph Traversal • Problem: Search for a certain node or traverse all nodes in the graph • Depth First Traversal – Once a possible path is found, continue the search until the end of the path • Breadth First Traversal – Start several paths at a time, and advance in each one step at a time
5. 5. Depth First traversal:  When a graph is traversed by visiting the nodes in the forward (deeper) direction as long as possible, the traversal is called depth-first traversal. For example, for the graph shown in Figure, the depth- first traversal starting at the vertex 0 visits the node in the orders:  0 1 2 6 7 8 5 3 4  0 4 3 5 8 6 7 2 1 Breadth first traversal  When a graph is traversed by visiting all the adjacent nodes/vertices of a node/vertex first, the traversal is called breadth-first traversal. For example, for a graph in which the breadth-first traversal starts at vertex v1, visits to the nodes take place in the order shown in Figure