Discrete Math with SageMath: Learn math with open-source software

A digraph, or directed graph, is a visual representation of a relation $R$ on the set $A\text{.}$ Every element in set $A$ is shown as a node (vertex). An arrow from the node $a$ to the node $b$ represents the pair $(a,b)$ on the relation $R\text{.}$

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# Define the set A

A = Set([2, 3, 4, 6, 8])

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# Define the relation R on A: aRb iff a divides b

R = [(a, b) for a in A for b in A if a.divides(b)]

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DiGraph(R, loops=true)

The circles intersecting the node are the same as arrows pointing from the node to itself.

We can add a title to the digraph with the name parameter.

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DiGraph(R, loops=true, name="Look at my digraph")

If the digraph does not contain a relation from a node to itself, we can omit the loops=true parameter. If we happen to forgot to include the parameter when we need to, Sage will give us a descriptive error message.

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# Define the set A

A = Set([2, 3, 4, 6, 8])

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# Define the relation R on A: aRb iff a < b

R = [(a, b) for a in A for b in A if a < b]

DiGraph(R)

We can also define the digraph using pair notion for relations.

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DiGraph([(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)])

Alternatively, we can define the digraph directly. The element on the left of the : is a node. The node relates to the elements in the list on the right of the :.

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# 1 relates to 2, 3, and 4

# 2 relates to 3 and 4

# 3 relates to 4

DiGraph({1: [2, 3, 4], 2: [3, 4], 3: [4]})