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# Adjacency Matrix

#### In this tutorial, you will learn what an adjacency matrix is. Also, you will find working examples of adjacency matrix in Python.

An adjacency matrix is a way of representing a graph G = {V, E} as a matrix of booleans.

## Adjacency matrix representation

The size of the matrix is `VxV`

where `V`

is the number of vertices in the graph and the value of an entry `Aij`

is either 1 or 0 depending on whether there is an edge from vertex i to vertex j.

## Adjacency Matrix Example

The image below shows a graph and its equivalent adjacency matrix.

In case of undirected graphs, the matrix is symmetric about the diagonal because of every edge `(i,j)`

, there is also an edge `(j,i)`

.

## Pros of adjacency matrix

The basic operations like adding an edge, removing an edge and checking whether there is an edge from vertex i to vertex j are extremely time efficient, constant time operations.

If the graph is dense and the number of edges is large, adjacency matrix should be the first choice. Even if the graph and the adjacency matrix is sparse, we can represent it using data structures for sparse matrices.

The biggest advantage however, comes from the use of matrices. The recent advances in hardware enable us to perform even expensive matrix operations on the GPU.

By performing operations on the adjacent matrix, we can get important insights into the nature of the graph and the relationship between its vertices.

## Cons of adjacency matrix

The `VxV`

space requirement of the adjacency matrix makes it a memory hog. Graphs out in the wild usually don’t have too many connections and this is the major reason why adjacency lists are the better choice for most tasks.

While basic operations are easy, operations like `inEdges`

and `outEdges`

are expensive when using the adjacency matrix representation.

## Python Examples

If you know how to create two dimensional arrays, you also know how to create an adjacency matrix.

```
/* Adjacency Matrix representation in Python */
class Graph(object):
/* Initialize the matrix */
def __init__(self, size):
self.adjMatrix = []
for i in range(size):
self.adjMatrix.append([0 for i in range(size)])
self.size = size
/* Add edges */
def add_edge(self, v1, v2):
if v1 == v2:
print("Same vertex %d and %d" % (v1, v2))
self.adjMatrix[v1][v2] = 1
self.adjMatrix[v2][v1] = 1
/* Remove edges */
def remove_edge(self, v1, v2):
if self.adjMatrix[v1][v2] == 0:
print("No edge between %d and %d" % (v1, v2))
return
self.adjMatrix[v1][v2] = 0
self.adjMatrix[v2][v1] = 0
def __len__(self):
return self.size
/* Print the matrix */
def print_matrix(self):
for row in self.adjMatrix:
for val in row:
print('{:4}'.format(val)),
print
def main():
g = Graph(5)
g.add_edge(0, 1)
g.add_edge(0, 2)
g.add_edge(1, 2)
g.add_edge(2, 0)
g.add_edge(2, 3)
g.print_matrix()
if __name__ == '__main__':
main()
```

## Adjacency Matrix Applications

- Creating routing table in networks
- Navigation tasks

# Python Example for Beginners

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