On certain regular nicely distance-balanced graphs

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