On certain regular nicely distance-balanced graphs

A connected graph $\G$ is called {\em nicely distance--balanced}, whenever there exists a positive integer $\gamma=\gamma(\G)$, such that for any two adjacent vertices $u,v$ of are exactly $\gamma$ which closer to $u$ than $v$, and $v$ $u$. Let $d$ denote the diameter $\G$. It known $d \le \gamma$, distance-balanced graphs with $\gamma = d$ precisely complete cycles length $2d$ or $2d+1$. In this paper we classify regular $\gamma=d+1$.