User blog:Hyp cos/SCG(n) and some related

I've obtained something about subcubic graph numbers. First, here are ordinal level of some graphs. A graph with larger ordinal isn't homeomorphically embeddable into any graph with smaller ordinal.

Here I use [something] to present a connected graph, and in the "something", I use numbers to present vertices, use m-n to present an edge between vertix m and n. For example, [1] is a dot, [1-1] is a dot with a loop, [1-2,1-2] is 2 dots with 2 edges connecting them, and [1-2,2-3,1-3] is a triangle.

Up to \(\varepsilon_0\)
It's a little different from tree(n) ordinal levels up to \(\varepsilon_0\). A dot has level 0

[1-2] has level 1

[1-2,2-3] has level 2

[1-2,2-3,3-4] has level 3

[1-2,1-3,1-4] has level \(\omega\)

[1-2,1-3,1-4,4-5] has level \(\omega+1\)

[1-2,1-3,1-4,4-5,5-6] has level \(\omega+2\)

[1-2,1-3,3-4,1-5,5-6] has level \(\omega2\)

[1-2,1-3,3-4,4-5,1-6,6-7,7-8] has level \(\omega3\)

[1-2,1-3,1-4,4-5,4-6] has level \(\omega^2\)

[1-2,1-3,1-4,4-5,4-6,6-7] has level \(\omega^2+1\)

[1-2,1-3,3-4,1-5,5-6,5-7,7-8] has level \(\omega^2+\omega\)

[1-2,1-3,1-4,4-5,5-6,5-7] has level \(\omega^22\)

[1-2,1-3,1-4,4-5,5-6,6-7,6-8] has level \(\omega^23\)

[1-2,1-3,1-4,4-5,4-6,6-7,6-8] has level \(\omega^3\)

[1-2,1-3,1-4,4-5,4-6,6-7,6-8,8-9,8-10] has level \(\omega^4\)

[1-2,2-3,1-4,4-5,1-6,6-7] has level \(\omega^\omega\)

[1-2,2-3,1-4,4-5,1-6,6-7,7-8] has level \(\omega^\omega+1\)

[1-2,2-3,1-4,4-5,1-6,6-7,6-8] has level \(\omega^\omega+\omega\)

[1-2,2-3,1-4,4-5,1-6,6-7,6-8,8-9,8-10] has level \(\omega^\omega+\omega^2\)

[1-2,2-3,1-4,4-5,5-6,1-7,7-8,8-9] has level \(\omega^\omega2\)

[1-2,2-3,1-4,4-5,4-6,1-7,7-8,7-9] has level \(\omega^{\omega+1}\)

[1-2,2-3,1-4,4-5,4-6,6-7,6-8,1-9,9-10,9-11,11-12,11-13] has level \(\omega^{\omega+2}\)

[1-2,2-3,1-4,4-5,1-6,6-7,7-8,6-9,9-10] has level \(\omega^{\omega2}\)

[1-2,2-3,1-4,4-5,1-6,6-7,7-8,8-9,7-10,10-11] has level \(\omega^{\omega2}2\)

[1-2,2-3,1-4,4-5,1-6,6-7,6-8,8-9,9-10,8-11,11-12] has level \(\omega^{\omega2+1}\)

[1-2,2-3,1-4,4-5,1-6,6-7,7-8,6-9,9-10,10-11,9-12,12-13] has level \(\omega^{\omega3}\)

[1-2,2-3,3-4,1-5,5-6,6-7,1-8,8-9,9-10] has level \(\omega^{\omega^2}\)

[1-2,2-3,3-4,4-5,1-6,6-7,7-8,8-9,1-10,10-11,11-12,12-13] has level \(\omega^{\omega^3}\)

[1-2,2-3,2-4,1-5,5-6,5-7,1-8,8-9,8-10] has level \(\omega^{\omega^\omega}\)