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## 7975Example: Examples of Adjacency Matrices

### A Digraph Example

The following figure shows a digraph $$D$$ with $$6$$ vertices and some edges:

This digraph has the adjacency matrix
$$\begin{array}{cccccccc} & & a & b & c & d & e & f \cr & & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \cr a & \rightarrow & 0 & 1 & 1 & 1 & 0 & 0 \cr b & \rightarrow & 2 & 0 & 0 & 0 & 0 & 0 \cr c & \rightarrow & 0 & 3 & 0 & 0 & 0 & 0 \cr d & \rightarrow & 0 & 0 & 0 & 0 & 2 & 0 \cr e & \rightarrow & 0 & 0 & 0 & 0 & 1 & 0 \cr f & \rightarrow & 0 & 0 & 0 & 0 & 0 & 0 \cr \end{array}$$

• is in general not symmetric,
• diagonal elements $$\neq 0$$ indicate loops,
• elements of $$> 1$$ indicate multiple edges.

### A Graph Example

The figure below demonstrates a similar graph with $$G$$ with $$6$$ vertices and some edges:

The adjacency matrix of this graph is given by

$$\begin{array}{cccccccc} & & a & b & c & d & e & f \cr & & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \cr a & \rightarrow & 0 & 3 & 1 & 1 & 0 & 0 \cr b & \rightarrow & 3 & 0 & 3 & 0 & 0 & 0 \cr c & \rightarrow & 1 & 3 & 0 & 0 & 0 & 0 \cr d & \rightarrow & 1 & 0 & 0 & 0 & 2 & 0 \cr e & \rightarrow & 0 & 0 & 0 & 2 & 1 & 0 \cr f & \rightarrow & 0 & 0 & 0 & 0 & 0 & 0 \cr \end{array}$$

• always symmetric,
• diagonal elements $$\neq 0$$ indicate loops,
• elements of $$> 1$$ indicate multiple edges.

| | | | Contributors: bookofproofs | References: [570]

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