(r, 2, r(r − 1))-regular graphs

A graph $G$ is called $( r , 2, r ( r - 1) )$-regular if each vertex in the graph $G$ is at a distance one away from exactly $r$ number of vertices and at a distance two away from exactly $r ( r - 1 )$ number of vertices. That is, $d(v) = r$ and $d_2 (v) = r (r-1 )$, for all $v$ in $G$. In this paper, we prove that for any $r > 0$, $r$-regular graph with girth at least five is $(r, 2, r(r-1))$-regular and vice versa and also suggest a method to construct $(r, 2, r(r-1))$-regular graph on $ n times 2 ^{r-2}$ vertices.