A tour of Archimedes' stomachion by Fan Chung and Ron Graham
What are interesting properties of the STOMACH graph?
The STOMACH graph has 268 vertices and 937 edges.
There are 936 edges in the giant component of the STOMACH graph and there is one minor
component with one edge.
The degree distribution is (0,3,6,1,8,51,100,77,18,2) for the giant component.
In other words, there are no vertices of degree 1, 3 of degree 2, 6 of degree 3, and so on.
Can you find the unique vertex of degree 4?
The diameter of the giant component of the STOMACH graph is 11.
Can you find pairs of tilings that are furthest away from each other?
For the giant component, the average distance is 4.810299... = 169539/35245.
For the supergraph of the original Stomachion tilings,
it has 17152 vertices. It has a giant component with 17024 vertices and a minor component
with 128 vertices.
The diameter of the giant component is 15 and the average distance is 6.716549.
The diameter of the minor component is 4.
We have computed the Laplacian spectrum of the STOMACH graph.
The induced subgraph on the giant component is Hamiltonian.
How many different Hamiltonian cycles are there for the giant component of STOMACH?
The number of spanning trees of the giant component of the STOMACH graph is
What is the number of spanning trees in the giant component of
the supergraph with 17024 vertices?
It is about 5.4 ... x 1020903.
The actual number, that we computed
in the spirit of the sandreckoner,
is too large to be written here.
To write down all the digits, it takes 8 pages in a word document.
We use the matrix-tree theorem, plus a few tricks, to find this number.