![Representation Of Graph
Two most common ways to represent a Graph:
1. Adjacency Matrix
2. Adjacency List
Adjacency Matrix:
Adjacency Matrix is a 2D array of size V x V where V is the number of
vertices in a graph.
Let the 2D array be adj [][], a slot adj [i][j] = 1 indicates that there is
an edge from vertex i to vertex j.
Adjacency matrix for undirected graph is always symmetric.
Adjacency Matrix is also used to represent weighted graphs.
If adj [i][j] = w, then there is an edge from vertex i to vertex j with
weight w.](https://image.slidesharecdn.com/graphtheory-190319062009/85/Graph-theory-8-320.jpg)
![Representation Of Graph
Adjacency List:
An array of lists is used.
Size of the array is equal to the number of vertices.
Let the array be array []. An entry array [i] represents the
list of vertices adjacent to the ith vertex.
This representation can also be used to represent a
weighted graph.
The weights of edges can be represented as lists of pairs.](https://image.slidesharecdn.com/graphtheory-190319062009/85/Graph-theory-9-320.jpg)
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Graph theory
- 2. Introduction Graph Theory is the study of points and lines. In particular, it involves the ways in which sets of points, called vertices, can be connected by lines called edges. Graph Theory has proven useful in the design of integrated circuits for computers and other electronic devices. Graph Theory was invented by Leonhard Euler.
- 3. Graphs And Basic Terminologies A Graph is an ordered pair G = (V, E) comprising a set V of vertices, nodes or points together with a set E of edges, arcs or lines, which are 2-element subsets of V. An edge is associated with two vertices, and the association takes the form of the unordered pair of the vertices. A Graph may be directed or undirected, weighted or unweighted. A Graph may be with self-loop or self-edge.
- 4. Example Of Graphs (a)Undirected Graph a b c d a b c 3 2 6 4 (b)Weighted Graph a bc (c)Directed Graph (d)Graph with self-loop a be cd
- 5. Types Of Graphs Null Graph: A Graph which contains only isolated nodes(i.e. set of edges of graph is empty) is called as Null Graph. Regular Graph: A Graph in which each vertex has the same number of neighbours(i.e. every vertex has the same degree) is called as Regular Graph. A Regular Graph with vertices of degree K is called a K -Regular Graph or regular graph of degree K. Complete Graph: A Complete Graph is a graph in which each pair of vertices is joined by an edge. A Complete Graph contains all possible edges.
- 6. Types Of Graphs Bipartite Graph: A Bipartite Graph is a graph in which the vertex set can be partitioned into two sets, W and X, so that no two vertices in W share a common edge and no two vertices in X share a common edge. Chromatic number of Bipartite Graph is 2. Complete Bipartite Graph: In a Complete Bipartite Graph, the vertex set is the union of two disjoint sets, W and X, so that every vertex in W is adjacent to every vertex in X but there are no edges within W or X.
- 7. Example Of Types Of Graphs (a)Null Graph With Three Vertices. a b c (b)Complete Graph With Three Vertices. a b cc db a (c)2-Regular Graph With Four Vertices. a b c ed (d)Bipartite Graph With Five Vertices. a b c d d e (e)k(3,3) Complete Bipartite Graph With Six Vertices.
- 8. Representation Of Graph Two most common ways to represent a Graph: 1. Adjacency Matrix 2. Adjacency List Adjacency Matrix: Adjacency Matrix is a 2D array of size V x V where V is the number of vertices in a graph. Let the 2D array be adj [][], a slot adj [i][j] = 1 indicates that there is an edge from vertex i to vertex j. Adjacency matrix for undirected graph is always symmetric. Adjacency Matrix is also used to represent weighted graphs. If adj [i][j] = w, then there is an edge from vertex i to vertex j with weight w.
- 9. Representation Of Graph Adjacency List: An array of lists is used. Size of the array is equal to the number of vertices. Let the array be array []. An entry array [i] represents the list of vertices adjacent to the ith vertex. This representation can also be used to represent a weighted graph. The weights of edges can be represented as lists of pairs.
- 10. Representation Of Graph Example Of Adjacency Matrix Of A Graph: a 1 2 3 1 0 1 1 2 1 0 1 b c 3 1 1 0 (a) Matrix Representation Of (a) Example Of Adjacency List Of A Graph: a b c (a) Head Node Vertex Node 1 2 3 NULL 2 1 3 NULL 3 1 2 NULL
- 11. Graph Algorithms Some Basic Graph Algorithms Are: 1.Depth First Search(DFS) 2.Breadth First Search(BFS) 3.Shortest Path Algorithm 4.Dijskstra’s Algorithm
- 12. Graph Coloring Graph Coloring is a method to assign colors to the vertices of a graph so that no two adjacent vertices have the same color. Some Graph Coloring problems are : Vertex coloring : A way of coloring the vertices of a graph so that no two adjacent vertices share the same color. Edge Coloring : It is the method of assigning a color to each edge so that no two adjacent edges have the same color. Face coloring : It assigns a color to each face or region of a planar graph so that no two faces that share a common boundary have the same color. Chromatic Number: Chromatic Number is the minimum number of colors required to color a graph.
- 13. THANK YOU Insert Group Member Names Here..