Number of nodes in a given distance in (random) regular graph Given a d-regular graph $G=<V,E>$ (connected, unweighted & simple), and a node $v$. 
denote all nodes with distance $k$ from $v$  $$L_k=\{u\in V : dis(v,u) = k\}$$
Let's call it "the k-th layer", where the distance $dis$ is taken as the length of the minimal path.
so, for instance, $L_0 = \{v\}$ and $|L_1|=d$.
In general: What is the size of the k-th layer? specifically, what is the size of the largest layer?
Do we know the answer under any interesting assumption?
e.g. 


*

*Do we know the expected answer for uniformly random d-regular graph for constant $d$ and $n\rightarrow\infty$ ?

*If we take $v \in V$ uniformly at random?

*If we take $v$ as the central node or The peripheral node?

*other assumptions?
 A: The magic words are "expander graph". A random regular graph is an expander, which means that the size of the layers is expanding until half the vertices are consumed.  This (more or less) answers your questions 1, 2. As for 3, I have no idea what central/peripheral node means.
A: This question is actually equivalent to the distribution of shortest path lengths in random graphs - Mor Nitzan, Eytan Katzav, Reimer Kühn, and Ofer Biham, Distance distribution in configuration-model networks, Phys. Rev. E 93, 062309 (2016).
In the case of random regular graphs there is an exact result, which is a Gompertz distribution. The tail distribution is given by
$$P(L> \ell) = \exp \left[ - \eta \left( e^{b \ell} - 1 \right) \right]$$
with $\eta=\frac{c}{(c-2)(N-1)}$ and $b=\ln(c-1)$
and where $c$ is the degree of all nodes (a random regular graph)
You can get the probability mass function via
$$P(L=\ell) = P(L > \ell - 1) - P(L > \ell)$$
and the expected number of nodes in each shell is simply $N \times P(L=\ell)$
Hope this helps
