# Graph and its representations

Graph is a data structure that consists of 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 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. 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 the 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.
Please see this for a sample Python implementation of adjacency matrix.

Adjacency List:
An array of lists is used. Size of the array is equal to the number of vertices. Let the array be 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 adjacency list representation of the above graph. Note that in below implementation, we use dynamic arrays (vector in C++/ArrayList in Java) to represent adjacency lists instead of linked list. The vector implementation has advantages of cache friendliness.

## C++

 `// A simple representation of graph using STL ` `#include ` `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 << ````" Adjacency list of vertex "``` `             ``<< v << ````" head "````; ` `        ``for` `(``auto` `x : adj[v]) ` `           ``cout << ``"-> "` `<< x; ` `        ``printf``(````" "````); ` `    ``} ` `} ` ` `  `// 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; ` `} `

## C

 `// A C Program to demonstrate adjacency list  ` `// representation of graphs ` `#include ` `#include ` ` `  `// 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 begining ` `    ``struct` `AdjListNode* newNode = newAdjListNode(dest); ` `    ``newNode->next = graph->array[src].head; ` `    ``graph->array[src].head = newNode; ` ` `  `    ``// Since graph is undirected, add an edge from ` `    ``// dest to src also ` `    ``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``(````" Adjacency list of vertex %d head "````, v); ` `        ``while` `(pCrawl) ` `        ``{ ` `            ``printf``(``"-> %d"``, pCrawl->dest); ` `            ``pCrawl = pCrawl->next; ` `        ``} ` `        ``printf``(````" "````); ` `    ``} ` `} ` ` `  `// Driver program to test above functions ` `int` `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

 `// Java Program to demonstrate adjacency list  ` `// representation of graphs ` `import` `java.util.LinkedList; ` ` `  `public` `class` `GFG  ` `{ ` `    ``// A user define class to represent a graph. ` `    ``// A graph is an array of adjacency lists. ` `    ``// Size of array will be V (number of vertices  ` `    ``// in graph) ` `    ``static` `class` `Graph ` `    ``{ ` `        ``int` `V; ` `        ``LinkedList adjListArray[]; ` `         `  `        ``// constructor  ` `        ``Graph(``int` `V) ` `        ``{ ` `            ``this``.V = V; ` `             `  `            ``// define the size of array as  ` `            ``// number of vertices ` `            ``adjListArray = ``new` `LinkedList[V]; ` `             `  `            ``// Create a new list for each vertex ` `            ``// such that adjacent nodes can be stored ` `            ``for``(``int` `i = ``0``; i < V ; i++){ ` `                ``adjListArray[i] = ``new` `LinkedList<>(); ` `            ``} ` `        ``} ` `    ``} ` `     `  `    ``// Adds an edge to an undirected graph ` `    ``static` `void` `addEdge(Graph graph, ``int` `src, ``int` `dest) ` `    ``{ ` `        ``// Add an edge from src to dest.  ` `        ``graph.adjListArray[src].add(dest); ` `         `  `        ``// Since graph is undirected, add an edge from dest ` `        ``// to src also ` `        ``graph.adjListArray[dest].add(src); ` `    ``} ` `      `  `    ``// A utility function to print the adjacency list  ` `    ``// representation of graph ` `    ``static` `void` `printGraph(Graph graph) ` `    ``{        ` `        ``for``(``int` `v = ``0``; v < graph.V; v++) ` `        ``{ ` `            ``System.out.println(``"Adjacency list of vertex "``+ v); ` `            ``System.out.print(``"head"``); ` `            ``for``(Integer pCrawl: graph.adjListArray[v]){ ` `                ``System.out.print(``" -> "``+pCrawl); ` `            ``} ` `            ``System.out.println(````" "````); ` `        ``} ` `    ``} ` `      `  `    ``// Driver program to test above functions ` `    ``public` `static` `void` `main(String args[]) ` `    ``{ ` `        ``// create the graph given in above figure ` `        ``int` `V = ``5``; ` `        ``Graph graph = ``new` `Graph(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); ` `    ``} ` `} ` `// This code is contributed by Sumit Ghosh `

## 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 {} head"````.``format``(i), end``=``"") ` `            ``temp ``=` `self``.graph[i] ` `            ``while` `temp: ` `                ``print``(``" -> {}"``.``format``(temp.vertex), end``=``"") ` `                ``temp ``=` `temp.``next` `            ``print``(````" "````) ` ` `  ` `  `# 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 `

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
```

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.

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

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.

This article is attributed to GeeksforGeeks.org

## tags:

Graph graph-basics Graph

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