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Arbitrary in my use: any node can be connected to any number of other
nodes. (assuming it's on the node-and-connection sort of style, which
it need not be if it has a fusional morphology)

I still don't see how you can say it's 6. Could you please show me the math?

I'm not sure about cases where 'neighbors' are directly touching, but
it's obviously false if they can be simply connected by lines. E.g.:

[paste this into a notepad w/ an equal-width font]

1 2 3
 \|/
4-O-5
 /|\
6 7 8

... and that's not even using multiple 'layers', or variable length
connecting lines, branching ones, or anything like that.

 - Sai