Compare adjacency matrices and adjacency lists as a graph changes, with synchronized views, interactive controls, and memory insights.
One graph · three synchronized views
Edit the matrix directly, then see each cell become an edge in the graph and an entry in the adjacency list.
Representation console
Create vertices, edit edges, then inspect how storage changes.
Vertices
Edge editor
Traversal lab
Click any matrix cell to add or remove its edge.
O(V²) storage · O(1) lookup
Click a cell to toggle its edge · diagonal cells represent self-loops
Visual graph
O(V + E) storage
Hover entries to locate their matrix cell and graph edge
The graph is dense, constant-time edge checks matter, or a compact 2D representation makes an algorithm easier to reason about. Its cost is always V² cells, even when most contain zero.
The graph is sparse or algorithms frequently iterate through neighbors. Storage grows with V + E, while checking one specific edge may require scanning a vertex’s neighbors.
Concept guide
Review the mental model, tradeoffs, and practical use cases after you experiment.
Two fundamental ways to represent graphs with different trade-offs. Adjacency Matrix uses a 2D array for O(1) edge lookups but O(V²) space, while Adjacency List uses arrays of lists for O(V + E) space but slower edge queries. Choice depends on graph density and operation frequency.
V × V matrix required
Constant time access
Resize matrix required
Update single cell
Stores all possible edges
Many edges per vertex
Vertices + edges storage
Scan adjacency list
Add to array/map
Append to list
Only stores existing edges
Few edges per vertex
| Operation | Adjacency Matrix | Adjacency List |
|---|---|---|
| Check Edge Existence | O(1) - matrix[i][j] | O(degree) - scan list |
| Get All Neighbors | O(V) - scan row | O(degree) - direct access |
| Add Vertex | O(V²) - resize matrix | O(1) - add to dictionary |
| Remove Vertex | O(V²) - resize matrix | O(E) - remove from all lists |
| Add Edge | O(1) - set cell value | O(1) - append to list |
| Remove Edge | O(1) - clear cell | O(degree) - search and remove |
| Language | Matrix | List |
|---|---|---|
| Python | [[0]*V for _ in range(V)] | {i: [] for i in range(V)} |
| Java | int[][] adj = new int[V][V] | List<List<Integer>> adj = new ArrayList<>() |
| C++ | vector<vector<int>> adj(V, vector<int>(V)) | vector<vector<int>> adj(V) |
| JavaScript | Array(V).fill().map(() => Array(V).fill(0)) | Array(V).fill().map(() => []) |
Sparse connections, many users
→ Adjacency List
Dense connections, fixed structure
→ Adjacency Matrix
Billions of pages, few links
→ Adjacency List
Many components, dense connections
→ Adjacency Matrix
Focus on existing connections
→ Adjacency List
| Type | Implementation | Space Complexity |
|---|---|---|
| Weighted Matrix | Store weights in cells, 0/-1 for no edge | O(V²) - same as unweighted |
| Weighted List | Store (neighbor, weight) pairs | O(V + E) - efficient for sparse |
| Directed Graphs | Asymmetric matrix |
