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
  
  4274907879584887999622870350791651653580561946098663 0260463080635888162932544968163511332884012787865253 8493848334747280548950296625028870945082815106322630 656323533502421487505199816146669473794347182850048
  = 4.27...x 10²⁰⁶.
  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.
  
  
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