Skip to main content

Adjacency matrix representation in Graph

Pictorially it is very the graph can be represented very easily and analysis can be done quite normally. But there is no direct pictorial data structure available to represent the graph. The basic data structure serve the purpose to store the graph in the memory. The data structures two-dimensional array and linked list most commonly used to represent the graphs. If the graph is represented using two dimensional array then the representational is called as "adjacency matrix representation" and if the linked list is used to represent the graph then the representation is called as "adjacency list representation".

Adjacency Matrix Representation:

A two dimensional array of size nxn can be used to represent the graph where 'n' is the number of vertices of the graph. 'n' number of rows are used to represent the vertices and 'n' number of columns are used for each vertex. The matrix entries depend on the type of graph. If the graph is undirected or directed then each row for the respective vertex contains the number of direct paths to each vertex is entered in the matrix. If the graph is weighted graph then the matrix entries for each row will be the weight of the edge to other vertices is entered as matrix entry. Consider the following undirected graph:

Adjacency matrix representation in Graph


In the given graph, the node 2, 4 and 6 are adjacent to 1(there exist a direct path). So, in the adjacency matrix the row entry for vertex 1 contains '1' for each column representing vertices 2, 4 and 6 and rest of the entries will be 0 for that row. Similarly the row for vertex 2 contains entry '1' for column 1 and 5, '0' for 2, 4 and 6. You can note that 1 and 5 are adjacent nodes of vertex 2. In our all discussions node and vertex are synonyms. The complete adjacency matrix is:

Adjacency matrix representation in Graph


In the above adjacency matrix each row is used for each node and each column is used for each node. Adj ij entry will be 1 if node 'i' is adjacent to 'j' otherwise it is 0 where 'Adj' is adjacency matrix, 'i' represents rows for nodes 1,2,3,4,5,6 and 'j' represents columns for nodes 1,2,4,5,6.

Consider the following digraph:

directed graph with adjacency

Nodes 2 and 6 are adjacent nodes of 1. Node 5 is the only adjacent node of 2. Nodes 1 and 5 are adjacent node 4. Node 6 is the adjacent node of 5. There are no adjacent nodes of node 6 it means it is not possible to move from node 6 to any other nodes of the graph (no direct path). So, the adjacency matrix is:

Adjacency matrix representation in Graph


The adjacency matrix of a directed graph is not always symmetric whereas the adjacency matrix of an undirected graph is always symmetric. So, determination of only upper triangle of the adjacency matrix is more than enough for an undirected graph.

Consider the following weighted graph:

Consider the following weighted graph

In the adjacency matrix of the above weighted graph the entries are the respective weight of the edges of the adjacent nodes if any exists otherwise it will be '∞' (unknown weight). So, the adjacency matrix is:
Adjacency matrix representation in Graph

Comments

Popular posts from this blog

difference between structure and union in C Language

In c language article we will see the difference between union and structure. Both are the user define datatype in c language. See the table which is mentioned below: ASP.NET Video Tutorial Series Structure Union1.The keywordstruct is used to define a structure 1. The keyword union is used to define a union. 2. When a variable is associated with a structure, the compiler allocates the memory for each member. The size of structure is greater than or equal to the sum ofsizes of its members. The smaller members may end with unused slack bytes. 2. When a variable is associated with a union, thecompiler allocates thememory by considering the size of the largest memory. So, size of union is equal to the size of largest member. 3. Each member within a structure is assigned unique storage area of location. 3. Memory allocated is shared by individual members of union. 4. The address of each member will be in ascending order This indicates that memory for each member will start at different offset v…

Difference between Linear search and Binary Search in c language

SQL Video Channel : Download all SQL Video



Binary Search Linear Search Works only on sorted items. such as  1,2,3,4,5,6  etc
Works on sorted as well as unsorted items. 12,4,5,3,2,1 etc Very efficient if the items are sorted Very efficient if the items are less and present in the beginning of the list. such as Suppose your list items are : 12,3,4,5,1 and you want to search 12 number then you get beginning in the list. Works well with arrays and not on linked lists. Works with arrays and linked lists.
Number of comparisons are less More number of comparisons are required if the items are present in the later part of the array or its elements are more.

Memory representation of Linked List Data Structures in C Language

Memory representation of Linked List

             In memory the linked list is stored in scattered cells (locations).The memory for each node is allocated dynamically means as and when required. So the Linked List can increase as per the user wish and the size is not fixed, it can vary.

               Suppose first node of linked list is allocated with an address 1008. Its graphical representation looks like the figure shown below:


      Suppose next node is allocated at an address 506, so the list becomes,



  Suppose next node is allocated with an address with an address 10,s the list become,


The other way to represent the linked list is as shown below:




 In the above representation the data stored in the linked list is “INDIA”, the information part of each node contains one character. The external pointer root points to first node’s address 1005. The link part of the node containing information I contains 1007, the address of next node. The last node …