Adjacency matrix

Definition

Let $G$ be a finite graph with vertex set $\{v_1, \dots, v_n\}$. The adjacency matrix of $G$ is the $n \times n$ matrix $A = (a_{ij})$ with $$a_{ij} = \begin{cases} 1 & \text{if } v_i \sim v_j, \\ 0 & \text{otherwise.} \end{cases}$$

Remarks

The matrix $A$ is symmetric and has zeros on the diagonal. The sum of row $i$ is the degree of $v_i$. The entry $(A^k)_{ij}$ is the number of walks of length $k$ from $v_i$ to $v_j$. Another order of the vertices gives the matrix $P^{T} A P$ for a permutation matrix $P$.

Explanations

L1 human

The adjacency matrix is a table of zeros and ones. A $1$ in row $i$ and column $j$ means that vertices $i$ and $j$ are joined. For a triangle, every entry off the diagonal is $1$.