Graph and its representations

Graph is a data structure that consists of the following two components: 

• A finite set of vertices also called nodes.
• A finite set of ordered pair of the form (u, v) called edge. The pair is ordered because (u, v) is not the same as (v, u) in the case of a directed graph(di-graph). The pair of the form (u, v) indicates that there is an edge from vertex u to vertex v. The edges may contain weight/value/cost.

Following is an example of an undirected graph with 5 vertices.

Example of undirected graph with 5 vertices

Graphs are used to represent many real-life applications: Graphs are used to represent networks. The networks may include paths in a city or telephone network or circuit network. Graphs are also used in social networks like linkedIn, Facebook. For example, in Facebook, each person is represented with a vertex(or node). Each node is a structure and contains information like person id, name, gender, and locale. See this for more applications of graph.

In computer science, a graph is a data structure that is used to represent connections or relationships between objects. A graph consists of a set of vertices (also known as nodes) and a set of edges (also known as arcs) that connect the vertices. The vertices can represent anything from cities in a map to web pages in a network, and the edges can represent the relationships between them, such as roads or links.

A graph can be visualized as a collection of points (vertices) connected by lines (edges), where each vertex represents a point of interest and each edge represents a connection between two points. The edges can be directed or undirected, meaning they can either have a specific direction or be bidirectional. For example, a map of a city may have directed edges that represent the direction of one-way streets, while a social network may have undirected edges that represent friendships between individuals.

Representations of Graphs:

The following two are the most commonly used representations of a graph.

• Adjacency Matrix
• Adjacency List

There are other representations also like, Incidence Matrix and Incidence List. The choice of graph representation is situation-specific. It totally depends on the type of operations to be performed and the ease of use.

Adjacency List:
An adjacency list is a simple way to represent a graph as a list of vertices, where each vertex has a list of its adjacent vertices. Here's an example of an adjacency list for an undirected graph with 4 vertices:
makefile
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0: 1 3
1: 0 2
2: 1 3
3: 0 2
In this example, vertex 0 is adjacent to vertices 1 and 3, vertex 1 is adjacent to vertices 0 and 2, and so on.

Adjacency Matrix:
An adjacency matrix is a two-dimensional array that represents the graph by storing a 1 at position (i,j) if there is an edge from vertex i to vertex j, and 0 otherwise. Here's an example of an adjacency matrix for the same undirected graph:
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  0 1 2 3
0 0 1 0 1
1 1 0 1 0
2 0 1 0 1
3 1 0 1 0
In this example, there is an edge from vertex 0 to vertex 1 (represented by a 1 in position (0,1)), an edge from vertex 1 to vertex 0 (represented by a 1 in position (1,0)), and so on.

Incidence Matrix:
An incidence matrix is a two-dimensional array that represents the graph by storing a 1 at position (i,j) if vertex i is incident on edge j, and 0 otherwise. Here's an example of an incidence matrix for the same undirected graph:
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  0 1 2 3
0 1 1 0 1
1 1 0 1 0
2 0 1 1 0
3 1 0 0 1
In this example, vertex 0 is incident on edges 0, 1, and 3 (represented by a 1 in positions (0,0), (0,1), and (0,3)), vertex 1 is incident on edges 0, 2 (represented by a 1 in positions (1,0) and (1,2)), and so on.

Each representation has its own advantages and disadvantages depending on the application, and choosing the right representation can have a significant impact on the performance of graph algorithms.

Adjacency Matrix:

Adjacency Matrix is a 2D array of size V x V where V is the number of vertices in a graph. Let the 2D array be adj[][], a slot adj[i][j] = 1 indicates that there is an edge from vertex i to vertex j. The adjacency matrix for an undirected graph is always symmetric.

Adjacency Matrix is also used to represent weighted graphs. If adj[i][j] = w, then there is an edge from vertex i to vertex j with weight w. 

We follow the below pattern to use the adjacency matrix in code:

• In the case of an undirected graph, we need to show that there is an edge from vertex i to vertex j and vice versa. In code, we assign adj[i][j] = 1  and adj[j][i] = 1.
• In the case of a directed graph, if there is an edge from vertex i to vertex j then we just assign adj[i][j]=1.\

See the undirected graph shown below:

Example of undirected graph with 5 vertices

The adjacency matrix for the above example graph is:

Adjacency matrix representation

Advantages of Adjacency Matrix:

• Representation is easier to implement and follow. 
• Removing an edge takes O(1) time. 
• Queries like whether there is an edge from vertex ‘u’ to vertex ‘v’ are efficient and can be done O(1).

Disadvantages of Adjacency Matrix:

• Consumes more space O(V²). Even if the graph is sparse(contains less number of edges), it consumes the same space.
• Adding a vertex takes O(V²) time. Computing all neighbors of a vertex takes O(V) time (Not efficient).

Implementation of Adjacency Matrix:

C

#include <stdio.h>

int main()
{
    // n is the number of vertices
    // m is the number of edges
    int n, m;
    scanf("%d %d", &n, &m);
    int adjMat[n + 1][n + 1];
    for (int i = 0; i < m; i++) {
        int u, v;
        scanf("%d %d", &u, &v);
        adjMat[u][v] = 1;
        adjMat[v][u] = 1;
        // for a directed graph with an edge pointing from u
        // to v,we just assign adjMat[u][v] as 1
    }

    return 0;
}

C++

#include <bits/stdc++.h>
using namespace std;

int main()
{
    // n is the number of vertices
    // m is the number of edges
    int n, m;
    cin >> n >> m;
    int adjMat[n + 1][n + 1];
    for (int i = 0; i < m; i++) {
        int u, v;
        cin >> u >> v;
        adjMat[u][v] = 1;
        adjMat[v][u] = 1;
        // for a directed graph with an edge pointing from u
        // to v,we just assign adjMat[u][v] as 1
    }

    return 0;
}

Java

import java.util.*;

public class Main {
    public static void main(String[] args)
    {
        Scanner sc = new Scanner(System.in);

        // n is the number of vertices
        // m is the number of edges
        int n = sc.nextInt();
        int m = sc.nextInt();
        int[][] adjMat = new int[n + 1][n + 1];
        for (int i = 0; i < m; i++) {
            int u = sc.nextInt();
            int v = sc.nextInt();
            adjMat[u][v] = 1;
            adjMat[v][u] = 1;
            // for a directed graph with an edge pointing
            // from u to v,we just assign adjMat[u][v] as 1
        }
    }
}

// This code is contributed by Tapesh(tapeshdua420)

Python3

if __name__ == '__main__':

    #  n is the number of vertices
    #  m is the number of edges
    n, m = map(int, input().split())
    adjMat = [[0 for i in range(n)]for j in range(n)]
    for i in range(n):
        u, v = map(int, input().split())
        adjMat[u][v] = 1
        adjMat[v][u] = 1
        # for a directed graph with an edge
        # pointing from u to v,we just assign
        # adjMat[u][v] as 1

C#

using System;
using System.IO;

class Program {
    static void Main(string[] args)
    {
        // n is the number of vertices
        // m is the number of edges
        int n, m;
        n = Convert.ToInt32(Console.ReadLine());
        m = Convert.ToInt32(Console.ReadLine());

        int[, ] adjMat = new int[n + 1, n + 1];

        for (int i = 0; i < m; i++) {
            int u, v;
            u = Convert.ToInt32(Console.ReadLine());
            v = Convert.ToInt32(Console.ReadLine());

            adjMat[u, v] = 1;
            adjMat[v, u] = 1;
            // for a directed graph with an edge pointing
            // from u to v,we just assign adjMat[u][v] as 1
        }
    }
}

// This code is contributed by Tapesh(tapeshdua420)

Javascript

var n;
    n = prompt();
    var m;
    m = prompt();

    var adjMat = new Array(n+1);
    for(let i = 0; i < adjMat.length; i++){
        adjMat[i] = new Array(n+1);
    }

    for(let i = 0; i < m; i++){
        var u = prompt();
        var v = prompt();
        adjMat[u][v] = 1;
        adjMat[v][u] = 1;
        // for a directed graph with an edge pointing from u to v,we just assign
        // adjMat[u][v] as 1
    }

    // This code is contributed by lokesh.

Adjacency List:

An array of linked lists is used. The size of the array is equal to the number of vertices. Let the array be an array[]. An entry array[i] represents the linked list of vertices adjacent to the i^th vertex.

This representation can also be used to represent a weighted graph. The weights of edges can be represented as lists of pairs.

Consider the following graph:

Example of undirected graph with 5 vertices

Following is the adjacency list representation of the above graph.

Adjacency List representation of the above graph

Advantages of Adjacency List:

• Saves space. Space taken is O(|V|+|E|). In the worst case, there can be C(V, 2) number of edges in a graph thus consuming O(V²) space.
• Adding a vertex is easier.
• Computing all neighbors of a vertex takes optimal time.

Disadvantages of Adjacency List:

Queries like whether there is an edge from vertex u to vertex v are not efficient and can be done O(V).

Implementation of Adjacency List:

Note that in the below implementation, we use dynamic arrays (vector in C++/ArrayList in Java) to represent adjacency lists instead of the linked list. The vector implementation has the advantage of cache friendliness. 

C

// A C Program to demonstrate adjacency list
// representation of graphs
#include <stdio.h>
#include <stdlib.h>

// A structure to represent an adjacency list node
struct AdjListNode {
    int dest;
    struct AdjListNode* next;
};

// A structure to represent an adjacency list
struct AdjList {
    struct AdjListNode* head;
};

// A structure to represent a graph. A graph
// is an array of adjacency lists.
// Size of array will be V (number of vertices
// in graph)
struct Graph {
    int V;
    struct AdjList* array;
};

// A utility function to create a new adjacency list node
struct AdjListNode* newAdjListNode(int dest)
{
    struct AdjListNode* newNode
        = (struct AdjListNode*)malloc(
            sizeof(struct AdjListNode));
    newNode->dest = dest;
    newNode->next = NULL;
    return newNode;
}

// A utility function that creates a graph of V vertices
struct Graph* createGraph(int V)
{
    struct Graph* graph
        = (struct Graph*)malloc(sizeof(struct Graph));
    graph->V = V;

    // Create an array of adjacency lists.  Size of
    // array will be V
    graph->array = (struct AdjList*)malloc(
        V * sizeof(struct AdjList));

    // Initialize each adjacency list as empty by
    // making head as NULL
    int i;
    for (i = 0; i < V; ++i)
        graph->array[i].head = NULL;

    return graph;
}

// Adds an edge to an undirected graph
void addEdge(struct Graph* graph, int src, int dest)
{
    // Add an edge from src to dest.  A new node is
    // added to the adjacency list of src.  The node
    // is added at the beginning
    struct AdjListNode* check = NULL;
    struct AdjListNode* newNode = newAdjListNode(dest);

    if (graph->array[src].head == NULL) {
        newNode->next = graph->array[src].head;
        graph->array[src].head = newNode;
    }
    else {

        check = graph->array[src].head;
        while (check->next != NULL) {
            check = check->next;
        }
        // graph->array[src].head = newNode;
        check->next = newNode;
    }

    // Since graph is undirected, add an edge from
    // dest to src also
    newNode = newAdjListNode(src);
    if (graph->array[dest].head == NULL) {
        newNode->next = graph->array[dest].head;
        graph->array[dest].head = newNode;
    }
    else {
        check = graph->array[dest].head;
        while (check->next != NULL) {
            check = check->next;
        }
        check->next = newNode;
    }

    // newNode = newAdjListNode(src);
    // newNode->next = graph->array[dest].head;
    // graph->array[dest].head = newNode;
}

// A utility function to print the adjacency list
// representation of graph
void printGraph(struct Graph* graph)
{
    int v;
    for (v = 0; v < graph->V; ++v) {
        struct AdjListNode* pCrawl = graph->array[v].head;
        printf("\n Adjacency list of vertex %d\n head ", v);
        while (pCrawl) {
            printf("-> %d", pCrawl->dest);
            pCrawl = pCrawl->next;
        }
        printf("\n");
    }
}

// Driver program to test above functions
int main()
{
    // create the graph given in above figure
    int V = 5;
    struct Graph* graph = createGraph(V);
    addEdge(graph, 0, 1);
    addEdge(graph, 0, 4);
    addEdge(graph, 1, 2);
    addEdge(graph, 1, 3);
    addEdge(graph, 1, 4);
    addEdge(graph, 2, 3);
    addEdge(graph, 3, 4);

    // print the adjacency list representation of the above
    // graph
    printGraph(graph);

    return 0;
}

C++

// A simple representation of graph using STL
#include <bits/stdc++.h>
using namespace std;

// A utility function to add an edge in an
// undirected graph.
void addEdge(vector<int> adj[], int u, int v)
{
    adj[u].push_back(v);
    adj[v].push_back(u);
}

// A utility function to print the adjacency list
// representation of graph
void printGraph(vector<int> adj[], int V)
{
    for (int v = 0; v < V; ++v) {
        cout << "\n Adjacency list of vertex " << v
             << "\n head ";
        for (auto x : adj[v])
            cout << "-> " << x;
        cout << endl;
    }
}

// Driver code
int main()
{
    int V = 5;
    vector<int> adj[V];
    addEdge(adj, 0, 1);
    addEdge(adj, 0, 4);
    addEdge(adj, 1, 2);
    addEdge(adj, 1, 3);
    addEdge(adj, 1, 4);
    addEdge(adj, 2, 3);
    addEdge(adj, 3, 4);
    printGraph(adj, V);
    return 0;
}

Java

// Java code to demonstrate Graph representation
// using ArrayList in Java

import java.util.*;

class Graph {

    // A utility function to add an edge in an
    // undirected graph
    static void addEdge(ArrayList<ArrayList<Integer> > adj,
                        int u, int v)
    {
        adj.get(u).add(v);
        adj.get(v).add(u);
        // for a directed graph with an edge pointing from u
        // to v, adj.get(u).add(v);
    }

    // A utility function to print the adjacency list
    // representation of graph
    static void
    printGraph(ArrayList<ArrayList<Integer> > adj)
    {
        for (int i = 0; i < adj.size(); i++) {
            System.out.println("\nAdjacency list of vertex"
                               + i);
            System.out.print("head");
            for (int j = 0; j < adj.get(i).size(); j++) {
                System.out.print(" -> "
                                 + adj.get(i).get(j));
            }
            System.out.println();
        }
    }

    // Driver Code
    public static void main(String[] args)
    {
        // Creating a graph with 5 vertices
        int V = 5;
        ArrayList<ArrayList<Integer> > adj
            = new ArrayList<ArrayList<Integer> >(V);

        for (int i = 0; i < V; i++)
            adj.add(new ArrayList<Integer>());

        // Adding edges one by one
        addEdge(adj, 0, 1);
        addEdge(adj, 0, 4);
        addEdge(adj, 1, 2);
        addEdge(adj, 1, 3);
        addEdge(adj, 1, 4);
        addEdge(adj, 2, 3);
        addEdge(adj, 3, 4);

        printGraph(adj);
    }
}

Python3

"""
A Python program to demonstrate the adjacency
list representation of the graph
"""


# A class to represent the adjacency list of the node
class AdjNode:
    def __init__(self, data):
        self.vertex = data
        self.next = None


# A class to represent a graph. A graph
# is the list of the adjacency lists.
# Size of the array will be the no. of the
# vertices "V"
class Graph:
    def __init__(self, vertices):
        self.V = vertices
        self.graph = [None] * self.V

    # Function to add an edge in an undirected graph
    def add_edge(self, src, dest):
        # Adding the node to the source node
        node = AdjNode(dest)
        node.next = self.graph[src]
        self.graph[src] = node

        # Adding the source node to the destination as
        # it is the undirected graph
        node = AdjNode(src)
        node.next = self.graph[dest]
        self.graph[dest] = node

    # Function to print the graph
    def print_graph(self):
        for i in range(self.V):
            print("Adjacency list of vertex {}\n head".format(i), end="")
            temp = self.graph[i]
            while temp:
                print(" -> {}".format(temp.vertex), end="")
                temp = temp.next
            print(" \n")


# Driver program to the above graph class
if __name__ == "__main__":
    V = 5
    graph = Graph(V)
    graph.add_edge(0, 1)
    graph.add_edge(0, 4)
    graph.add_edge(1, 2)
    graph.add_edge(1, 3)
    graph.add_edge(1, 4)
    graph.add_edge(2, 3)
    graph.add_edge(3, 4)

    graph.print_graph()

# This code is contributed by Kanav Malhotra

C#

// C# code to demonstrate Graph representation
// using LinkedList in C#
using System;
using System.Collections.Generic;

class Graph {
    // A utility function to add an edge in an
    // undirected graph
    static void addEdge(LinkedList<int>[] adj, int u, int v)
    {
        adj[u].AddLast(v);
        adj[v].AddLast(u);
    }

    // A utility function to print the adjacency list
    // representation of graph
    static void printGraph(LinkedList<int>[] adj)
    {
        for (int i = 0; i < adj.Length; i++) {
            Console.WriteLine("\nAdjacency list of vertex "
                              + i);
            Console.Write("head");

            foreach(var item in adj[i])
            {
                Console.Write(" -> " + item);
            }
            Console.WriteLine();
        }
    }

    // Driver Code
    public static void Main(String[] args)
    {
        // Creating a graph with 5 vertices
        int V = 5;
        LinkedList<int>[] adj = new LinkedList<int>[ V ];

        for (int i = 0; i < V; i++)
            adj[i] = new LinkedList<int>();

        // Adding edges one by one
        addEdge(adj, 0, 1);
        addEdge(adj, 0, 4);
        addEdge(adj, 1, 2);
        addEdge(adj, 1, 3);
        addEdge(adj, 1, 4);
        addEdge(adj, 2, 3);
        addEdge(adj, 3, 4);

        printGraph(adj);

        Console.ReadKey();
    }
}

// This code is contributed by techno2mahi

Javascript

<script>
// Javascript code to demonstrate Graph representation
// using ArrayList in Java

// A utility function to add an edge in an
    // undirected graph
function addEdge(adj,u,v)
{
    adj[u].push(v);
        adj[v].push(u);
}

// A utility function to print the adjacency list
    // representation of graph
function printGraph(adj)
{
    for (let i = 0; i < adj.length; i++) {
            document.write("<br>Adjacency list of vertex" + i+"<br>");
            document.write("head");
            for (let j = 0; j < adj[i].length; j++) {
                document.write(" -> "+adj[i][j]);
            }
            document.write("<br>");
        }
}

// Driver Code
// Creating a graph with 5 vertices
        let V = 5;
        let adj= [];

        for (let i = 0; i < V; i++)
            adj.push([]);

        // Adding edges one by one
        addEdge(adj, 0, 1);
        addEdge(adj, 0, 4);
        addEdge(adj, 1, 2);
        addEdge(adj, 1, 3);
        addEdge(adj, 1, 4);
        addEdge(adj, 2, 3);
        addEdge(adj, 3, 4);

        printGraph(adj);


// This code is contributed by avanitrachhadiya2155
</script>

Output

 Adjacency list of vertex 0
 head -> 1-> 4

 Adjacency list of vertex 1
 head -> 0-> 2-> 3-> 4

 Adjacency list of vertex 2
 head -> 1-> 3

 Adjacency list of vertex 3
 head -> 1-> 2-> 4

 Adjacency list of vertex 4
 head -> 0-> 1-> 3

Reference: 
http://en.wikipedia.org/wiki/Graph_%28abstract_data_type%29
Related Post:
Graph representation using STL for competitive programming | Set 1 (DFS of Unweighted and Undirected)
Graph implementation using STL for competitive programming | Set 2 (Weighted graph)
This article is compiled by Aashish Barnwal and reviewed by GeeksforGeeks team. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.