Notes

Chapter 9: Fundamental Physics

Section 7: Space as a Network


Trivalent networks

With n nodes and 3 connections at each node a network must always have an even number of nodes, and a total of 3n/2 connections. Of all possible such networks, most large ones end up being connected. The number of distinct such networks for even n from 2 to 10 is {2, 5, 17, 71, 388}. If no self connections are allowed then these numbers become {1, 2, 6, 20, 91}, while if neither self nor multiple connections are allowed (yielding what are often referred to as cubic or 3-regular graphs), the numbers become {0, 1, 2, 5, 19, 85, 509, 4060, 41301, 510489}, or asymptotically (6 n)! /( (3 n)! (2 n)! 288^n E^2). (For symmetric graphs see page 1032.) If one requires the networks to be planar the numbers are {0, 1, 1, 3, 9, 32, 133, 681, 3893, 24809, 169206}. If one looks at subnetworks with dangling connections, the number of these up to size 10 is {2, 5, 7, 22, 43, 141, 373, 1270, 4053, 14671}, or {1, 1, 2, 6, 10, 29, 64, 194, 531, 1733} if no self or multiple connections are allowed (see also page 1039).

From Stephen Wolfram: A New Kind of Science [citation]