GraphUtilities`
GraphUtilities`

# GraphDistance

As of Version 10, all the functionality of the GraphUtilities package is built into the Wolfram System. »

GraphDistance[g,i,j]

gives the distance from vertex i to vertex j in the graph g.

# Details and Options

• GraphDistance functionality is now available in the built-in Wolfram Language function GraphDistance.
• To use GraphDistance, you first need to load the Graph Utilities Package using Needs["GraphUtilities`"].
• GraphDistance returns the graph distance from one vertex to another. Infinity is returned if no path exists from i to j. By default every edge is assumed to have an edge weight of 1.
• The following option can be used:
•  Weighted False specifies whether edge weight is to be used in calculating distance

# Examples

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## Basic Examples(2)

This defines a simple directed graph:

This calculates the distance from vertex 1 to vertex 5:

This function has been superseded by GraphDistance in the Wolfram System:

## Scope(1)

This defines a simple directed graph:

This calculates the distance from vertex 1 to vertex 5:

There is no path from vertex 4 to vertex 1:

This defines a simple directed graph with edge weights:

This shows that the distance from vertex 1 to 3 is 2 if unit edge weights are assumed:

But the distance becomes 0 if edge weights are taken into account:

## Applications(1)

This generates a random directed graph with n nodes and degree d:

Average distance between vertices 1 and 100, of 200 random directed graphs of 100 vertices and degree 2:

The average distance decreases with an increase in the degree:

## Possible Issues(1)

This defines a small directed graph:

This shows that the distance between vertices 1 and 3 is 2, assuming a unit edge weight:

This shows that when edge weights are taken into account, the distance between vertices 1 and 3 is 0:

This shows the shortest path when edge weights are taken into account:

This shows the shortest path when edge weights are ignored:

Wolfram Research (2007), GraphDistance, Wolfram Language function, https://reference.wolfram.com/language/GraphUtilities/ref/GraphDistance.html.

#### Text

Wolfram Research (2007), GraphDistance, Wolfram Language function, https://reference.wolfram.com/language/GraphUtilities/ref/GraphDistance.html.

#### CMS

Wolfram Language. 2007. "GraphDistance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/GraphUtilities/ref/GraphDistance.html.

#### APA

Wolfram Language. (2007). GraphDistance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/GraphUtilities/ref/GraphDistance.html

#### BibTeX

@misc{reference.wolfram_2024_graphdistance, author="Wolfram Research", title="{GraphDistance}", year="2007", howpublished="\url{https://reference.wolfram.com/language/GraphUtilities/ref/GraphDistance.html}", note=[Accessed: 10-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_graphdistance, organization={Wolfram Research}, title={GraphDistance}, year={2007}, url={https://reference.wolfram.com/language/GraphUtilities/ref/GraphDistance.html}, note=[Accessed: 10-September-2024 ]}