A graph is a type of non-linear data structure made up of nodes and edges. The nodes are also known as vertices, and edges are lines or arcs that connect any two nodes in the graph.
The following are the two most common graph representations:
1. Adjacency Matrix: Adjacency Matrix is a two-dimensional array with the dimensions V x V, where V is the number of vertices in a graph. Representation is simpler to implement and adhere to. It takes O(1) time to remove an edge. Queries such as whether there is an edge from vertex 'u' to vertex 'v' are efficient and can be completed in O(1).
One of the cons of this representation is that even if the graph is sparse (has fewer edges), it takes up the same amount of space. Adding a vertex takes O(V^2). It also takes O(V) time to compute all of a vertex's neighbours, which is not very efficient.
2. Adjacency List: In this method, each Node holds a list of Nodes that are directly connected to that vertex. Each node at the end of the list is connected with null values to indicate that it is the last node in the list. This saves space O(|V|+|E|). In the worst-case scenario, a graph can have C(V, 2) edges, consuming O(V^2) space. It is simpler to add a vertex. It takes the least amount of time to compute all of a vertex's neighbours.
One of the cons of this representation is that queries such as "is there an edge from vertex u to vertex v?" are inefficient and take O (V) in the worst case.
