# 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.
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 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.

An array of linked 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 linked 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 stored in nodes of linked lists. Following is adjacency list representation of the above graph.

Below is the code for adjacency list representation of an undirected graph:

## 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.
{
}

// A utility function to print the adjacency list
// representation of graph
{
for (int v = 0; v < V; ++v)
{
cout << "\n Adjacency list of vertex "
<< v << "\n head ";
cout << "-> " << x;
printf("\n");
}
}

// Driver code
int main()
{
int V = 5;
return 0;
}
```

## 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
{
int dest;
};

// A structure to represent an adjacency list
{
};

// 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;
};

// A utility function to create a new adjacency list node
{
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 =

// Initialize each adjacency list as empty by
int i;
for (i = 0; i < V; ++i)

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
// is added at the begining

// Since graph is undirected, add an edge from
// dest to src also
}

// 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)
{
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 fugure
int V = 5;
struct Graph* graph = createGraph(V);

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

return 0;
}
```

## Java

```// Java Program to demonstrate adjacency list
// representation of graphs

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;

// constructor
Graph(int V)
{
this.V = V;

// define the size of array as
// number of vertices

// Create a new list for each vertex
// such that adjacent nodes can be stored
for(int i = 0; i < V ; i++){
}
}
}

// Adds an edge to an undirected graph
static void addEdge(Graph graph, int src, int dest)
{
// Add an edge from src to dest.

// Since graph is undirected, add an edge from dest
// to src also
}

// 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(" -> "+pCrawl);
}
System.out.println("\n");
}
}

// 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);

// print the adjacency list representation of
// the above graph
printGraph(graph);
}
}
// This code is contributed by Sumit Ghosh
```

Output:

```

head -> 0-> 2-> 3-> 4

```

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).
The text passage describes the concept of graphs in computer science, their components, and different representations such as adjacency matrix and adjacency list. It includes code examples in C++, C, and Java for creating and printing an adjacency list representation of a graph.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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.
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 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.

An array of linked 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 linked 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 stored in nodes of linked lists. Following is adjacency list representation of the above graph.

Below is the code for adjacency list representation of an undirected graph:

## 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.
{
}

// A utility function to print the adjacency list
// representation of graph
{
for (int v = 0; v < V; ++v)
{
cout << "\n Adjacency list of vertex "
<< v << "\n head ";
cout << "-> " << x;
printf("\n");
}
}

// Driver code
int main()
{
int V = 5;
return 0;
}
```

## 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
{
int dest;
};

// A structure to represent an adjacency list
{
};

// 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;
};

// A utility function to create a new adjacency list node
{
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 =

// Initialize each adjacency list as empty by
int i;
for (i = 0; i < V; ++i)

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
// is added at the begining

// Since graph is undirected, add an edge from
// dest to src also
}

// 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)
{
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 fugure
int V = 5;
struct Graph* graph = createGraph(V);

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

return 0;
}
```

## Java

```// Java Program to demonstrate adjacency list
// representation of graphs

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;

// constructor
Graph(int V)
{
this.V = V;

// define the size of array as
// number of vertices

// Create a new list for each vertex
// such that adjacent nodes can be stored
for(int i = 0; i < V ; i++){
}
}
}

// Adds an edge to an undirected graph
static void addEdge(Graph graph, int src, int dest)
{
// Add an edge from src to dest.

// Since graph is undirected, add an edge from dest
// to src also
}

// 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(" -> "+pCrawl);
}
System.out.println("\n");
}
}

// 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);

// print the adjacency list representation of
// the above graph
printGraph(graph);
}
}
// This code is contributed by Sumit Ghosh
```

Output:

```

head -> 0-> 2-> 3-> 4

```

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).
user
The text passage describes the concept of graphs in computer science, their components, and different representations such as adjacency matrix and adjacency list. It includes code examples in C++, C, and Java for creating and printing an adjacency list representation of a graph.

TYPE: NO_QUESTION