Adjacency Matrix

Definition

Consider a Graph with $n$ nodes $X_1,X_2,\dots ,X_n$.

Unweighted Adjacency Matrix

An unweighted adjacency matrix $A$ is $n\times n$ matrix such that its element $a_{ij}$ is one when there is edge from node $X_i$ to $X_j$ and zero when there is no edge.

Weighted Adjacency Matrix

An unweighted adjacency matrix $A$ is $n\times n$ matrix such that its element $a_{ij}$ is the weight of the edge between $X_i$ and $X_j$, and is zero when there is no edge.

Examples

Given graph

Adjacency matrix $A=\left(\begin{array}{cccccc}2& 1& 0& 0& 1& 0\\ 1& 0& 1& 0& 1& 0\\ 0& 1& 0& 1& 0& 0\\ 0& 0& 1& 0& 1& 1\\ 1& 1& 0& 1& 0& 0\\ 0& 0& 0& 1& 0& 0\end{array}\right)$

Facts

If a graph is undirected, then the adjacency matrix is Symmetric Matrix