Adjacency Matrix

In an adjacency matrix m, the cell m[i][j] represents the edge weight from node i to node j.

[[0,0,2,0,3,-4],
[0,0,0,0,5,-2],
[0,0,0,6,9,0],
[0,0,0,0,0,0],
[0,0,0,0,0,8],
[0,0,0,0,0,0]]

The matrix represents the graph in the picture above.

Pros Cons
Space efficient for representing dense graphs Requires O(V2) space
Edge weight lookup is O(1) Iterating over all edges takes O(V2 time
Easy to understand  

Adjacency List

An adjacency list represents a graph as a map from nodes to list of edges. The first value in the tuple represents the to node and the second value represents the weight. It is most commonly used in solving graph problems.

0 -> [(2,2),(4,3),(5,-4)]
1 -> [(4,5),(5,-2)]
2 -> [(3,6),(4,9)]
3 -> []
4 -> [(5,8)]
5 -> []

Pros Cons
Space efficient for representing sparse graphs Less space efficient for denser graphs
Iterating over all edges is efficient Edge weight lookup is O(E)
  Slightly more difficult to understand

Edge List

An edge list is an unordered list of edges. The triplet (u,v,w) means the cost from node u to node v is weight w. It is seldom used because of its lack of structure.

[(2,3,6),
(2,4,9),
(0,2,2),
(0,4,3),
(0,5,-4),
(4,5,8),
(1,4,5),
(1,5,-2)]

Pros Cons
Space efficient for representing sparse graphs Less space efficient for denser graphs
Iterating over all edges is efficient Edge weight lookup is O(E)
Very simple structure