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?