Complexities of Graph Operations
Let's discuss the performance of the two graph representation approaches.
We'll cover the following
Time Complexities #
Below, you can find the time complexities for the 4 basic graph functions.
Note, that in this table V means the total number of vertices and E means the total number of edges in the Graph.
Operation  Adjacency List  Adjacency Matrix 

Add Vertex  O(1)  O(V^{2}) 
Remove Vertex  O(V+E)  O(V^{2}) 
Add Edge  O(1)  O(1) 
Remove Edge  O(E)  O(1) 
Adjacency List #

Addition operations in adjacency lists take constant time as we only need to insert at the head node of the corresponding vertex.

Removing an edge takes O(E) time, because in the worst case, all the edges could be at a single vertex and, hence, we would have to traverse all E edges to reach the last one.

Removing a vertex takes O(V + E) time because we have to delete all its edges and then reindex the rest of the list one step back in order to fill the deleted spot.
Adjacency Matrix #

Edge operations are performed in constant time as we only need to manipulate the value in the particular cell.

Vertex operations are performed in O(V^{2}) since we need to add rows and columns. We will also need to fill all the new cells.
Comparison #
Both representations are suitable for different situations. If your model frequently manipulates vertices, the adjacency list is a better choice.
If you are dealing primarily with edges, the adjacency matrix is the more efficient approach.
Keep these complexities in mind because they will give you a better idea about the time complexities of the several algorithms we’ll see in this section.
In the next lesson, we will look at a special type of graph called the bipartite graph.
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