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<!DOCTYPE html>

<head>

<title>Graph Adjacency Matrix </title>

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<h3>Page Index</h3>

<ul>

<li><a href="/dsa.html#data-structure-1" title="Data Structures (I)">Data Structures

(I)</a></li>

<li><a href="/dsa.html#data-structure-2" title="Data Structures (II)">Data Structures

(II)</a></li>

<li><a href="/dsa.html#tree-1" title="Tree based DSA (I)">Tree based DSA (I)</a></li>

<li><a href="/dsa.html#tree-2" title="Tree based DSA (II)">Tree based DSA (II)</a></li>

<li><a href="/dsa.html#graph" title="Graph Data Structures and Algorithm">Graph based

DSA</a></li>

<li><a href="/dsa.html#sorting-searching" title="Sorting and Searching">Sorting and

Searching</a></li>

<li><a href="/dsa.html#greedy-algorithm" title="Greedy Algorithms">Greedy Algorithms</a>

</li>

<li><a href="/dsa.html#dynamic-programming" title="Dynamic Programming">Dynamic

Programming</a></li>

<li><a href="/dsa.html#other-algorithms" title="Other Algorithms">Other Algorithms</a>

</li>

</ul>

</div>

<h1>Adjacency Matrix</h1>

<p class="editor-contents__short-description">In this tutorial, you will learn what an adjacency matrix is. Also, you will find working examples of adjacency matrix in C, C++, Java and Python.</p>

<p id="introduction">An adjacency matrix is a way of representing a graph as a matrix of booleans (0's and 1's). A finite graph can be represented in the form of a square matrix on a computer, where the boolean value of the matrix indicates if there is a direct path between two vertices.</p>

<p>For example, we have a graph below.</p>

<figure><img alt="An undirected graph" src="https://www.programiz.com/sites/tutorial2program/files/adjacency-matrix-graph.png" title="An undirected graph" width="242" height="242"><figcaption>An undirected graph</figcaption></figure><p>We can represent this graph in matrix form like below.</p>

<figure><img alt="Matrix representation of the graph" src="https://www.programiz.com/sites/tutorial2program/files/adjacency-matrix-representation_1.png" title="Matrix representation of the graph" width="365" height="302"><figcaption>Matrix representation of the graph</figcaption></figure><p>Each cell in the above table/matrix is represented as <code>A<sub>ij</sub></code>, where <code>i</code> and <code>j</code> are vertices. The value of <code>A<sub>ij</sub></code> is either 1 or 0 depending on whether there is an edge from vertex <code>i</code> to vertex <code>j</code>.</p>

<p></p><div class="clearfix"></div><p>If there is a path from <code>i</code> to <code>j</code>, then the value of <code>A<sub>ij</sub></code> is 1 otherwise its 0. For instance, there is a path from vertex 1 to vertex 2, so <code>A<sub>12</sub></code> is 1 and there is no path from vertex 1 to 3, so <code>A<sub>13</sub></code> is 0.</p>

<p>In case of undirected graphs, the matrix is symmetric about the diagonal because of every edge <code>(i,j)</code>, there is also an edge <code>(j,i)</code>.</p>

<hr><h2 id="pros">Pros of Adjacency Matrix</h2>

<ul><li>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.</li>

<li>If the graph is dense and the number of edges is large, an 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.</li>

<li>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.</li>

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

</ul><hr><h2 id="cons">Cons of Adjacency Matrix</h2>

<ul><li>The <code>VxV</code> 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 <a href="/dsa/graph-adjacency-list.html">adjacency lists</a> are the better choice for most tasks.</li>

<li>While basic operations are easy, operations like <code>inEdges</code> and <code>outEdges</code> are expensive when using the adjacency matrix representation.</li>

</ul><hr><h2 id="code">Adjacency Matrix Code in Python, Java, and C/C++</h2>

<p>If you know how to create two-dimensional arrays, you also know how to create an adjacency matrix.</p>

<div id='py1'class="tabbed-editor__node tabbed-editor__node--active" onclick="changepy()"><a href="#python-code">Python</a></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="python hljs"><span class="hljs-comment"># Adjacency Matrix representation in Python</span>

<span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">Graph</span><span class="hljs-params">(object)</span>:</span>

<span class="hljs-comment"># Initialize the matrix</span>

self.adjMatrix = []

<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> range(size):

self.adjMatrix.append([<span class="hljs-number">0</span> <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> range(size)])

self.size = size

<span class="hljs-comment"># Add edges</span>

<span class="hljs-keyword">if</span> v1 == v2:

<span class="hljs-keyword">print</span>(<span class="hljs-string">"Same vertex %d and %d"</span> % (v1, v2))

self.adjMatrix[v1][v2] = <span class="hljs-number">1</span>

self.adjMatrix[v2][v1] = <span class="hljs-number">1</span>

<span class="hljs-comment"># Remove edges</span>

<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">remove_edge</span><span class="hljs-params">(self, v1, v2)</span>:</span>

<span class="hljs-keyword">if</span> self.adjMatrix[v1][v2] == <span class="hljs-number">0</span>:

<span class="hljs-keyword">print</span>(<span class="hljs-string">"No edge between %d and %d"</span> % (v1, v2))

<span class="hljs-keyword">return</span>

self.adjMatrix[v1][v2] = <span class="hljs-number">0</span>

self.adjMatrix[v2][v1] = <span class="hljs-number">0</span>

<span class="hljs-keyword">return</span> self.size

<span class="hljs-comment"># Print the matrix</span>

<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">print_matrix</span><span class="hljs-params">(self)</span>:</span>

<span class="hljs-keyword">for</span> row <span class="hljs-keyword">in</span> self.adjMatrix:

<span class="hljs-keyword">for</span> val <span class="hljs-keyword">in</span> row:

<span class="hljs-keyword">print</span>(<span class="hljs-string">'{:4}'</span>.format(val)),

<span class="hljs-keyword">print</span>

g = Graph(<span class="hljs-number">5</span>)

g.add_edge(<span class="hljs-number">0</span>, <span class="hljs-number">1</span>)

g.add_edge(<span class="hljs-number">0</span>, <span class="hljs-number">2</span>)

g.add_edge(<span class="hljs-number">1</span>, <span class="hljs-number">2</span>)

g.add_edge(<span class="hljs-number">2</span>, <span class="hljs-number">0</span>)

g.add_edge(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>)

g.print_matrix()

<span class="hljs-keyword">if</span> __name__ == <span class="hljs-string">'__main__'</span>:

main()</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="java hljs"><span class="hljs-comment">// Adjacency Matrix representation in Java</span>

<span class="hljs-keyword">public</span> <span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">Graph</span> </span>{

<span class="hljs-keyword">private</span> <span class="hljs-keyword">boolean</span> adjMatrix[][];

<span class="hljs-keyword">private</span> <span class="hljs-keyword">int</span> numVertices;

<span class="hljs-comment">// Initialize the matrix</span>

<span class="hljs-function"><span class="hljs-keyword">public</span> <span class="hljs-title">Graph</span><span class="hljs-params">(<span class="hljs-keyword">int</span> numVertices)</span> </span>{

<span class="hljs-keyword">this</span>.numVertices = numVertices;

adjMatrix = <span class="hljs-keyword">new</span> <span class="hljs-keyword">boolean</span>[numVertices][numVertices];

}

<span class="hljs-comment">// Add edges</span>

<span class="hljs-function"><span class="hljs-keyword">public</span> <span class="hljs-keyword">void</span> <span class="hljs-title">addEdge</span><span class="hljs-params">(<span class="hljs-keyword">int</span> i, <span class="hljs-keyword">int</span> j)</span> </span>{

adjMatrix[i][j] = <span class="hljs-keyword">true</span>;

adjMatrix[j][i] = <span class="hljs-keyword">true</span>;

}

<span class="hljs-comment">// Remove edges</span>

<span class="hljs-function"><span class="hljs-keyword">public</span> <span class="hljs-keyword">void</span> <span class="hljs-title">removeEdge</span><span class="hljs-params">(<span class="hljs-keyword">int</span> i, <span class="hljs-keyword">int</span> j)</span> </span>{

adjMatrix[i][j] = <span class="hljs-keyword">false</span>;

adjMatrix[j][i] = <span class="hljs-keyword">false</span>;

}

<span class="hljs-comment">// Print the matrix</span>

<span class="hljs-function"><span class="hljs-keyword">public</span> String <span class="hljs-title">toString</span><span class="hljs-params">()</span> </span>{

StringBuilder s = <span class="hljs-keyword">new</span> StringBuilder();

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < numVertices; i++) {

s.append(i + <span class="hljs-string">": "</span>);

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">boolean</span> j : adjMatrix[i]) {

s.append((j ? <span class="hljs-number">1</span> : <span class="hljs-number">0</span>) + <span class="hljs-string">" "</span>);

}

s.append(<span class="hljs-string">"\n"</span>);

}

<span class="hljs-keyword">return</span> s.toString();

}

<span class="hljs-function"><span class="hljs-keyword">public</span> <span class="hljs-keyword">static</span> <span class="hljs-keyword">void</span> <span class="hljs-title">main</span><span class="hljs-params">(String args[])</span> </span>{

Graph g = <span class="hljs-keyword">new</span> Graph(<span class="hljs-number">4</span>);

g.addEdge(<span class="hljs-number">0</span>, <span class="hljs-number">1</span>);

g.addEdge(<span class="hljs-number">0</span>, <span class="hljs-number">2</span>);

g.addEdge(<span class="hljs-number">1</span>, <span class="hljs-number">2</span>);

g.addEdge(<span class="hljs-number">2</span>, <span class="hljs-number">0</span>);

g.addEdge(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>);

System.out.print(g.toString());

}

}</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="c hljs cpp"><span class="hljs-comment">// Adjacency Matrix representation in C</span>

<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><stdio.h></span></span>

<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> V 4</span>

<span class="hljs-comment">// Initialize the matrix to zero</span>

<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">init</span><span class="hljs-params">(<span class="hljs-keyword">int</span> arr[][V])</span> </span>{

<span class="hljs-keyword">int</span> i, j;

<span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i < V; i++)

<span class="hljs-keyword">for</span> (j = <span class="hljs-number">0</span>; j < V; j++)

arr[i][j] = <span class="hljs-number">0</span>;

}

<span class="hljs-comment">// Add edges</span>

<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">addEdge</span><span class="hljs-params">(<span class="hljs-keyword">int</span> arr[][V], <span class="hljs-keyword">int</span> i, <span class="hljs-keyword">int</span> j)</span> </span>{

arr[i][j] = <span class="hljs-number">1</span>;

arr[j][i] = <span class="hljs-number">1</span>;

}

<span class="hljs-comment">// Print the matrix</span>

<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">printAdjMatrix</span><span class="hljs-params">(<span class="hljs-keyword">int</span> arr[][V])</span> </span>{

<span class="hljs-keyword">int</span> i, j;

<span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i < V; i++) {

<span class="hljs-built_in">printf</span>(<span class="hljs-string">"%d: "</span>, i);

<span class="hljs-keyword">for</span> (j = <span class="hljs-number">0</span>; j < V; j++) {

<span class="hljs-built_in">printf</span>(<span class="hljs-string">"%d "</span>, arr[i][j]);

}

<span class="hljs-built_in">printf</span>(<span class="hljs-string">"\n"</span>);

}

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{

<span class="hljs-keyword">int</span> adjMatrix[V][V];

init(adjMatrix);

addEdge(adjMatrix, <span class="hljs-number">0</span>, <span class="hljs-number">1</span>);

addEdge(adjMatrix, <span class="hljs-number">0</span>, <span class="hljs-number">2</span>);

addEdge(adjMatrix, <span class="hljs-number">1</span>, <span class="hljs-number">2</span>);

addEdge(adjMatrix, <span class="hljs-number">2</span>, <span class="hljs-number">0</span>);

addEdge(adjMatrix, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>);

printAdjMatrix(adjMatrix);

<span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;

}</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="cpp hljs"><span class="hljs-comment">// Adjacency Matrix representation in C++</span>

<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><iostream></span></span>

<span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> <span class="hljs-built_in">std</span>;

<span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">Graph</span> {</span>

<span class="hljs-keyword">private</span>:

<span class="hljs-keyword">bool</span>** adjMatrix;

<span class="hljs-keyword">int</span> numVertices;

<span class="hljs-keyword">public</span>:

<span class="hljs-comment">// Initialize the matrix to zero</span>

Graph(<span class="hljs-keyword">int</span> numVertices) {

<span class="hljs-keyword">this</span>->numVertices = numVertices;

adjMatrix = <span class="hljs-keyword">new</span> <span class="hljs-keyword">bool</span>*[numVertices];

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < numVertices; i++) {

adjMatrix[i] = <span class="hljs-keyword">new</span> <span class="hljs-keyword">bool</span>[numVertices];

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> j = <span class="hljs-number">0</span>; j < numVertices; j++)

adjMatrix[i][j] = <span class="hljs-literal">false</span>;

}

<span class="hljs-comment">// Add edges</span>

<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">addEdge</span><span class="hljs-params">(<span class="hljs-keyword">int</span> i, <span class="hljs-keyword">int</span> j)</span> </span>{

adjMatrix[i][j] = <span class="hljs-literal">true</span>;

adjMatrix[j][i] = <span class="hljs-literal">true</span>;

}

<span class="hljs-comment">// Remove edges</span>

<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">removeEdge</span><span class="hljs-params">(<span class="hljs-keyword">int</span> i, <span class="hljs-keyword">int</span> j)</span> </span>{

adjMatrix[i][j] = <span class="hljs-literal">false</span>;

adjMatrix[j][i] = <span class="hljs-literal">false</span>;

}

<span class="hljs-comment">// Print the martix</span>

<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">toString</span><span class="hljs-params">()</span> </span>{

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < numVertices; i++) {

<span class="hljs-built_in">cout</span> << i << <span class="hljs-string">" : "</span>;

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> j = <span class="hljs-number">0</span>; j < numVertices; j++)

<span class="hljs-built_in">cout</span> << adjMatrix[i][j] << <span class="hljs-string">" "</span>;

<span class="hljs-built_in">cout</span> << <span class="hljs-string">"\n"</span>;

}

~Graph() {

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < numVertices; i++)

<span class="hljs-keyword">delete</span>[] adjMatrix[i];

<span class="hljs-keyword">delete</span>[] adjMatrix;

}

};

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{

<span class="hljs-function">Graph <span class="hljs-title">g</span><span class="hljs-params">(<span class="hljs-number">4</span>)</span></span>;

g.addEdge(<span class="hljs-number">0</span>, <span class="hljs-number">1</span>);

g.addEdge(<span class="hljs-number">0</span>, <span class="hljs-number">2</span>);

g.addEdge(<span class="hljs-number">1</span>, <span class="hljs-number">2</span>);

g.addEdge(<span class="hljs-number">2</span>, <span class="hljs-number">0</span>);

g.addEdge(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>);

g.toString();

}</code></pre></div>

</div>

<hr><h2 id="applications">Adjacency Matrix Applications</h2>

<ul><li>Creating routing table in networks</li>

<li>Navigation tasks</li>

</ul></div>

</div>

<div class="tutorial-toc"><div class="tutorial-toc__inner"><h3 class="tutorial-toc__title">Table of Contents

<li><a href="#pros">Pros of Adjacency Matrix</a></li>

<li><a href="#cons">Cons of Adjacency Matrix</a></li>

<li><a href="#code">Adjacency Matrix Code in Python, Java, and C/C++</a></li>

<li><a href="#applications">Adjacency Matrix Applications</a></li>

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Strongly Connected Components</a></button>

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Adjacency Matrix</a></button>

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Adjacency List</a></button>

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