BUILT-IN MATHEMATICA SYMBOL

VertexEccentricity

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

MORE INFORMATION

VertexEccentricity gives the vertex eccentricity for the connected component in which s is contained.

EXAMPLES

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Basic Examples (1)

Find the vertex eccentricity of vertex 1 in a graph:

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Out[1]=

In[2]:=

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Scope (3)

VertexEccentricity works with undirected graphs:

Directed graphs:

Large graphs:

Applications (6)

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 Barabasi-Albert random graph:

The de Solla Price random graph:

People at a family gathering:

Use Subsets to create edges between all members of a family:

The resulting social graph:

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

Build a network of related concepts based on the "See Also" section of reference pages:

These are the "See Also" entries for Graph:

Generate the whole "See Also" graph, with edges directed to the page where the "See Also" item appears:

Use VertexEccentricity to compare importance of concepts—the lower, the more important:

Properties & Relations (3)

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

GraphDistanceMatrix:

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

GraphRadius:

GraphPeriphery:

GraphCenter:

Illustrate 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: