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Tuesday, September 23, 2014
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Adjacency matrix representation in Graph

7:25 AMTuesday, September 23, 2014
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
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