Graph and its representations

07 May, 2022
A graph is a data structure that consists of the following two components: 1. A finite set of vertices also called as nodes. 2. A finite set of ordered pair of the form (u, v) called as edge. The pair is ordered because (u, v) is not the same as (v, u) in 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.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. Following is an example of an undirected graph with 5 vertices.  
The following two are the most commonly used representations of a graph. 1. Adjacency Matrix 2. 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 ease of use. 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. Adjacency matrix for 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. 
The adjacency matrix for the above example graph is:  
Pros: 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).Cons: Consumes more space O(V^2). Even if the graph is sparse(contains less number of edges), it consumes the same space. Adding a vertex is O(V^2) time.  Computing all neighbors of a vertex takes O(V) time (Not efficient).Please see this for a sample Python implementation of adjacency matrix.
Implementation of taking input for adjacency matrix
C++
#include <iostream>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 ;    }              return 0;}
Adjacency List: An array of 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 list of vertices adjacent to the ith vertex. This representation can also be used to represent a weighted graph. The weights of edges can be represented as lists of pairs. Following is the adjacency list representation of the above graph.  
 
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 advantages of cache friendliness.  
C++
C
Java
Python3
C#
Javascript
// 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 graphvoid 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;        printf("\n");    }} // Driver codeint 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;}
// A C Program to demonstrate adjacency list// representation of graphs#include <stdio.h>#include <stdlib.h> // A structure to represent an adjacency list nodestruct AdjListNode {    int dest;    struct AdjListNode* next;}; // A structure to represent an adjacency liststruct 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 nodestruct 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 verticesstruct 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 graphvoid 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 graphvoid 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 functionsint main(){    // create the graph given in above fugure    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;}
// 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);    }     // 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);    }}
"""A Python program to demonstrate the adjacencylist 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 classif __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# 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
<script>// Javascript code to demonstrate Graph representation// using ArrayList in Java // A utility function to add an edge in an    // undirected graphfunction addEdge(adj,u,v){    adj[u].push(v);        adj[v].push(u);} // A utility function to print the adjacency list    // representation of graphfunction 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>
 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
Pros: Saves space O(|V|+|E|) . In the worst case, there can be C(V, 2) number of edges in a graph thus consuming O(V^2) space. Adding a vertex is easier. Computing all neighbors of a vertex takes optimal time.Cons: Queries like whether there is an edge from vertex u to vertex v are not efficient and can be done O(V). In Real-life problems,  graphs are sparse(|E| <<|V|2). That’s why adjacency lists Data structure is commonly used for storing graphs. Adjacency matrix will enforce (|V|2) bound on time complexity for such algorithms. 
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Reference: http://en.wikipedia.org/wiki/Graph_%28abstract_data_type%29Related 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. 
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