What are Distance Regular Graphs

Question

I have been trying to understand distance regular graphs and how to compute the intersection array.

Distance Regular Graphs, this is the resource I have used. I could not figure what is br in δ(v,u)=r.br

Secondly cr in cr is the number of vertices that are adjacent to u and a distance of r − 1 from v

Lastly the two clauses for intersection array in the given link.

​ P.S : I need its concept to understand a paper, would appreciate the help

Answer 1

Quoting the paper, with some hopefully helpful comments in red.

Distance regular graphs have an intersection array $\{b_0,b_1,\ldots,b_{d−1};c_1,c_2, \ldots,c_d\}$ where for any two vertices $v$ and $u$ that are $r$ distance apart, $δ(v,u) = r$.$\color{red}{\leftarrow \text{a period.}}$ $b_r$ is the number of vertices that are adjacent to $u$ and at a distance $r + 1$ to $v$.$\color{red}{\leftarrow \text{another period.}}$ $c_r$ is the number of vertices that are adjacent to $u$ and a distance of $r − 1$ from $v$.

The definition should be clear now.