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

Please note that an adjacency matrix of a digraph

  • 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

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

Please note that an adjacency matrix of a graph is

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

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


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Bibliography (further reading)

[570] Krumke Sven O., Noltemeier Hartmut: “Graphentheoretische Konzepte und Algorithmen”, Teubner, 2005, Auflage 1

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