# adjacency-matrix representation

(data structure)

**Definition:**
A representation of a *directed graph* with n *vertices* using an n × n *matrix*, where the entry at (i,j) is 1 if there is an *edge* from vertex i to vertex j; otherwise the entry is 0. A *weighted graph* may be represented using the weight as the entry. An *undirected graph* may be represented using the same entry in both (i,j) and (j,i) or using an *upper triangular matrix*.

**Aggregate parent** (I am a part of or used in ...)

*graph*.

**See also**
*adjacency-list representation*, *dense graph*.

*Note:
Suppose we have a directed graph with four vertices. Here are adjacency-matrix and adjacency-list representations. The arrow (->) means a link in a list.
*

*
*

* 1 2 3 4*

1 1 1 1 1

2 1 0 0 0

3 0 1 0 1

4 0 1 1 0

*
*

1 -> 1 -> 2 -> 3 -> 4

2 -> 1

3 -> 2 -> 4

4 -> 2 -> 3

* The adjacency-list representation is more compact for a **sparse matrix*.

Author: SKS

## Implementation

Algorithms and Data Structures' implementation (Java and C++).

Go to the
Dictionary of Algorithms and Data
Structures home page.

If you have suggestions, corrections, or comments, please get in touch
with Paul Black.

Entry modified 3 August 2009.

HTML page formatted Tue Feb 12 10:57:42 2019.

Cite this as:

Sandeep Kumar Shukla, "adjacency-matrix representation", in
*Dictionary of Algorithms and Data Structures* [online], Vreda Pieterse and Paul E. Black, eds. 3 August 2009. (accessed TODAY)
Available from: https://www.nist.gov/dads/HTML/adjacencyMatrixRep.html