What is meant by diameter of a network?

The Wikipedia Example

Looks like the diameter is 3 to me by definition.

The longest shortest paths have length of 3 edges, e.g. between 6-1 and 6-2.

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The Mesh Example

Here’s your second definition, with some typographical correction so that it makes sense:

Diameter D of a network is defined as the longest path of the shortest paths between any two nodes. For example, the diameter of a 4×4 mesh D = 6

Let’s take a look at the 4×4 mesh example:

A---B---C---D
|   |   |   |
E---F---G---H
|   |   |   |
I---J---K---L
|   |   |   |
M---N---O---P

The longest shortest path has length of 6 edges, i.e. between A-P and M-D.

References

• Mathworld – Wolfram/Graph DiameterThe length of the “longest shortest path” between any two graph vertices of a graph.
• Graph and Digraph Glossary – cudenver.eduDiameter: The diameter of a graph is the length of the longest chain you are forced to use to get from one vertex to another in that graph. You can find the diameter of a graph by finding the distance between every pair of vertices and taking the maximum of those distances.

See also

• Computing Distances and Diameter
  • Has examples on weighted graphs