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:

Adjacency Matrix Representation

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.

Adjacency List Representation of Graph

""" 
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