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An **adjacency matrix** is a way to represent a graph as a **square table (matrix)**, where rows and columns correspond to vertices, and the cell values show whether there is an edge between them.

---

### Definition

For a graph with vertices ( V = {v_1, v_2, ..., v_n} ):
the adjacency matrix is a matrix ( A[n][n] ), where
[
A[i][j] =
\begin{cases}
1, & \text{if there is an edge from } v_i \text{ to } v_j, \
0, & \text{if there is no edge.}
\end{cases}
]

If the graph is **weighted**, the cell stores the **edge weight** instead of 1 and 0.

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### Example (undirected graph)

Edges: A-B, A-C, B-C

```javascript
    A B C
A [ 0 1 1 ]
B [ 1 0 1 ]
C [ 1 1 0 ]
```

The matrix is symmetric about the diagonal (A-B = B-A).

---

### Example (directed graph)

Edges: A→B, B→C

```javascript
    A B C
A [ 0 1 0 ]
B [ 0 0 1 ]
C [ 0 0 0 ]
```

---

### Features

- The matrix size is always **n × n**, where n is the number of vertices.
- **A[i][j] = 1** - there is a connection, **A[i][j] = 0** - there is none.
- In directed graphs, the matrix is **not symmetric**.
- A loop (an edge from a vertex to itself) is shown on the **diagonal**.

---

**Summary:**
an adjacency matrix is a table where, from the coordinates of two vertices, you can instantly find out whether there is an edge between them (and what its weight is).

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