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<!DOCTYPE html>
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<li><a href="/dsa.html#data-structure-1" title="Data Structures (I)">Data Structures
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<li><a href="/dsa.html#data-structure-2" title="Data Structures (II)">Data Structures
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<h1>Graph Data Stucture</h1>
<p class="editor-contents__short-description">In this tutorial, you will learn what a Graph Data Structure is. Also, you will find representations of a graph.</p>
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<p id="introduction">A graph data structure is a collection of nodes that have data and are connected to other nodes.</p>
<p>Let's try to understand this through an example. On facebook, everything is a node. That includes User, Photo, Album, Event, Group, Page, Comment, Story, Video, Link, Note...anything that has data is a node.</p>
<p>Every relationship is an edge from one node to another. Whether you post a photo, join a group, like a page, etc., a new edge is created for that relationship.</p>
<figure><img alt="graph data structure explained using facebook's example. Users, groups, pages, events, etc. are represented as nodes and their relationships - friend, joining a group, liking a page are represented as links between nodes" src="//cdn.programiz.com/sites/tutorial2program/files/facebook-graph.png" title="Example of graph data structure" width="730" height="352"><figcaption>Example of graph data structure</figcaption></figure><p>All of facebook is then a collection of these nodes and edges. This is because facebook uses a graph data structure to store its data.</p>
<p>More precisely, a graph is a data structure (V, E) that consists of</p>
<ul><li>A collection of vertices V</li>
<li>A collection of edges E, represented as ordered pairs of vertices (u,v)</li>
</ul><figure><img alt="a graph contains vertices that are like points and edges that connect the points" src="//cdn.programiz.com/sites/tutorial2program/files/graph-vertices-edges_0.png" title="Vertices and edges" width="730" height="242"><figcaption>Vertices and edges</figcaption></figure><p>In the graph,</p>
<pre>V = {0, 1, 2, 3}
E = {(0,1), (0,2), (0,3), (1,2)}
G = {V, E}</pre>
<hr><h2 id="term">Graph Terminology</h2>
<ul><li><strong>Adjacency</strong>: A vertex is said to be adjacent to another vertex if there is an edge connecting them. Vertices 2 and 3 are not adjacent because there is no edge between them.</li>
<li><strong>Path</strong>: A sequence of edges that allows you to go from vertex A to vertex B is called a path. 0-1, 1-2 and 0-2 are paths from vertex 0 to vertex 2.</li>
<li><strong>Directed Graph</strong>: A graph in which an edge (u,v) doesn't necessarily mean that there is an edge (v, u) as well. The edges in such a graph are represented by arrows to show the direction of the edge.</li>
</ul><hr><h2 id="representation">Graph Representation</h2>
<p>Graphs are commonly represented in two ways:</p>
<h3>1. Adjacency Matrix</h3>
<p>An adjacency matrix is a 2D array of V x V vertices. Each row and column represent a vertex.</p>
<p>If the value of any element <code>a[i][j]</code> is 1, it represents that there is an edge connecting vertex i and vertex j.</p>
<p></p><div class="clearfix"></div><p>The adjacency matrix for the graph we created above is</p>
<figure><img alt="graph adjacency matrix for sample graph shows that the value of matrix element is 1 for the row and column that have an edge and 0 for row and column that don't have an edge" src="//cdn.programiz.com/sites/tutorial2program/files/adjacency-matrix_1.png" title="Graph adjacency matrix" width="730" height="302"><figcaption>Graph adjacency matrix</figcaption></figure><p>Since it is an undirected graph, for edge (0,2), we also need to mark edge (2,0); making the adjacency matrix symmetric about the diagonal.</p>
<p>Edge lookup(checking if an edge exists between vertex A and vertex B) is extremely fast in adjacency matrix representation but we have to reserve space for every possible link between all vertices(V x V), so it requires more space.</p>
<h3>2. Adjacency List</h3>
<p>An adjacency list represents a graph as an array of linked lists.</p>
<p>The index of the array represents a vertex and each element in its linked list represents the other vertices that form an edge with the vertex.</p>
<p>The adjacency list for the graph we made in the first example is as follows:</p>
<figure><img alt="adjacency list representation represents graph as array of linked lists where index represents the vertex and each element in linked list represents the edges connected to that vertex" src="//cdn.programiz.com/sites/tutorial2program/files/adjacency-list.png" title="Adjacency list representation" width="730" height="304"><figcaption>Adjacency list representation</figcaption></figure><p>An adjacency list is efficient in terms of storage because we only need to store the values for the edges. For a graph with millions of vertices, this can mean a lot of saved space.</p>
<hr><h2 id="operation">Graph Operations</h2>
<p>The most common graph operations are:</p>
<ul><li>Check if the element is present in the graph</li>
<li>Graph Traversal</li>
<li>Add elements(vertex, edges) to graph</li>
<li>Finding the path from one vertex to another</li>
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<div class="tutorial-toc"><div class="tutorial-toc__inner"><h3 class="tutorial-toc__title">Table of Contents
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<li><a href="#term">Graph Terminology</a></li>
<li><a href="#representation">Graph Representation</a></li>
<li><a href="#operation">Graph Operations</a></li>
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