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Subject:

Date: Thu, 05 Aug 1993 16:55:36 -0700 From: "Ronnie B. Kon" <ronnie@cisco.com>

Disclaimer: this sounds more authoritative than is intended--I really

don't know what I'm talking about.

Don't worry. Mathematical reasoning stands on its own merits.

It couldn't be very pointy. From the most distant configuration,

there are 6 positions immediately before it. There are 6^2 two steps

away, 6^3 three steps, etc. (well, 6^2 - 1 and 6^3 - ?) actually.

Very good. This is a necessary insight, regardless of the exact

numerical details. (For example, you mean 12, not 6.) But the

possible flaw is that there might be more than one maximally distant

state; if their sets of neighbors overlap viciously enough, this

effect could make the tail pointier. You can make valence-12 graphs

(not of groups, just arbitrary graphs) that have fairly bumpy

distance-vs-population functions. Any argument that rigorously

constrains N(d) must somehow appeal to the fact that the cube graph is

a Cayley graph, that is, the graph of a group.

[...]

This gives me the feeling that Monte Carlo is fairly valid. (How's

that for rigor?)

It's a start. But we have to use groupness somehow.