# VertexEccentricity

VertexEccentricity[g,s]

gives the length of the longest shortest path from the source s to every other vertex in the graph g.

VertexEccentricity[{vw,},]

uses rules vw to specify the graph g.

# Details and Options

• VertexEccentricity is also known as node eccentricity.
• VertexEccentricity[g,s] gives the vertex eccentricity for the connected component in which s is contained.
• The following options can be given:
•  EdgeWeight Automatic weight for each edge Method Automatic method to use
• Possible Method settings include "BellmanFord" and "Dijkstra".

# Examples

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## Basic Examples(1)

Find the vertex eccentricity of vertex 1 in a graph:

## Scope(7)

VertexEccentricity works with undirected graphs:

Directed graphs:

Weighted graphs:

Multigraphs:

Mixed graphs:

Use rules to specify the graph:

VertexEccentricity works with large graphs:

## Applications(5)

In an PetersenGraph, every vertex has the same eccentricity:

Some Petersen graphs have different eccentricities for the inner and outer subgraphs:

Compute and highlight the vertex eccentricity for special graphs, including GridGraph:

Package this up as a function:

Many special graphs have constant vertex eccentricity:

A few will have varying eccentricity, where some vertices are more centrally located:

Most random graphs have small eccentricities:

The Gilbert random graph:

The BarabasiAlbert random graph:

The de Solla Price random graph:

Low eccentricity indicates close relation to everybody at the family gathering. Compare Larry and Rudy:

## Properties & Relations(3)

In a connected graph, the vertex eccentricity is related to GraphDistance:

The vertex eccentricity in a connected graph is related to GraphDiameter:

Illustrate the eccentricity of two vertices in a Petersen graph:

For a CompleteGraph, every vertex has eccentricity 1:

The eccentricity path in a PathGraph switches halfway through:

The eccentricity path in a CycleGraph measures both the GraphDiameter and GraphRadius:

In a WheelGraph of size 5 or more, the eccentricity is 1 at the hub and 2 elsewhere:

In a GridGraph, the eccentricity path always ends in a corner of the grid:

In a CompleteKaryTree, the eccentricity path always ends in a leaf:

Wolfram Research (2010), VertexEccentricity, Wolfram Language function, https://reference.wolfram.com/language/ref/VertexEccentricity.html (updated 2015).

#### Text

Wolfram Research (2010), VertexEccentricity, Wolfram Language function, https://reference.wolfram.com/language/ref/VertexEccentricity.html (updated 2015).

#### CMS

Wolfram Language. 2010. "VertexEccentricity." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/VertexEccentricity.html.

#### APA

Wolfram Language. (2010). VertexEccentricity. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/VertexEccentricity.html

#### BibTeX

@misc{reference.wolfram_2024_vertexeccentricity, author="Wolfram Research", title="{VertexEccentricity}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/VertexEccentricity.html}", note=[Accessed: 13-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_vertexeccentricity, organization={Wolfram Research}, title={VertexEccentricity}, year={2015}, url={https://reference.wolfram.com/language/ref/VertexEccentricity.html}, note=[Accessed: 13-July-2024 ]}