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.



Some questions about a family of regular undirected graphs
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](https://www.emis.de/journals/ELA/ela-articles/articles/vol17_pp154-167.pdf) for the Johnson graphs.


