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):
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.