When we look at how much time and space Breadth-First Search (BFS) and Depth-First Search (DFS) use, we need to think about how graphs are shown. There are two main ways: the adjacency matrix and the adjacency list.

Time Complexity:
The time it takes for BFS and DFS to run mostly depends on how the graph is represented and how many vertices (points) and edges (connections) there are.

1. Adjacency Matrix:

   • BFS: (O(V^2))
   • DFS: (O(V^2))

   In an adjacency matrix, checking if there is a connection between points takes constant time, or (O(1)). But to find out the connections, we have to look at all the vertices, making the total time (O(V^2)).

2. Adjacency List:

   • BFS: (O(V + E))
   • DFS: (O(V + E))

   With an adjacency list, we consider each point and its connections. Looking at each point takes (O(V)) time, and checking the edges adds another (O(E)) time.

Space Complexity:
The space that BFS and DFS use also depends on how the graph is set up.

1. Adjacency Matrix:

   • BFS: (O(V))
   • DFS: (O(V))

   For both algorithms, most of the space is used for keeping track of which points have been visited and, for BFS, a queue of points to examine. Since we can represent the graph with (V^2) space, it's manageable and still keeps it at (O(V)).

2. Adjacency List:

   • BFS: (O(V))
   • DFS: (O(V))

   An adjacency list needs (O(V + E)) to store the graph. However, the extra space used for BFS and DFS, like the queue or the stack of points, remains (O(V)).

In summary, both BFS and DFS have the same time complexity of (O(V + E)) when using adjacency lists. But if we use adjacency matrices, the time complexity can go up to (O(V^2)). For space, both methods use (O(V)) efficiently in all formats. This means the way we represent the graph is very important for how well BFS and DFS work.