Let $S$ be the regular n-dimensional simplex. We create a graph of where the vertices are m-faces and two vertices are connected if there exist a common (m-1)-face which they share. Then the number of vertices is $binomial(n,m)$ and the graph $G$ is regular with $m(n-m)$ vertices.

Prove or disprove:

1) The diameter of $G$ is $diam(G)=min(n-m,m)$

2) $G$ is periodic in the sence of Chris Godsil (google: periodic graph chris godsil)

3) $G$ is distance regular in the sence of Wikipedia.

Also there seems to be a connection to "brains" as follows:

a) Human brain:

1) $n = 205, m = 5$ , $binomial(n,m)\equiv 10^9=$ number of neurons

2) $m(m-n) \equiv 1000 = $ number of synapses per neuron

3) $diam(G)=min(205-5,5)=5 \equiv 6$ = diameter of $G$.

b) Brain of Ciona intestinalis (Wikipedia):

1) $n=22,m=2$

2) $binomial(n,m)=231$

3) $n(n-m)=40$

4) $diam(G)=2\equiv 3$

Thanks for you help in proving or disproving the conjectures above. What I have tried so far is writing a python program to produce the graphs and conjecture about the points above.

  • 4
    $\begingroup$ These are just the Johnson graphs. So maybe google that? $\endgroup$ – David Roberson Dec 3 '18 at 22:16

As for your question 2), according to https://arxiv.org/pdf/0806.2074.pdf, a regular graph is periodic if and only if its eigenvalues are distinct, which is the case for the Johnson graphs.


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