On distance-balanced generalized Petersen graphs

A connected graph $G$ of diameter ${\rm diam}(G) \ge \ell$ is $\ell$-distance-balanced if $|W_{xy}|=|W_{yx}|$ for every $x,y\in V(G)$ with $d_{G}(x,y)=\ell$, where $W_{xy}$ is the set of vertices of $G$ that are closer to $x$ than to $y$. We prove that the generalized Petersen graph $GP(n,k)$ is ${\rm diam}(GP(n,k))$-distance-balanced provided that $n$ is large enough relative to $k$. This partially solves a conjecture posed by Miklavi\v{c} and \v{S}parl \cite{Miklavic:2018}. We also determine ${\rm diam}(GP(n,k))$ when $n$ is large enough relative to $k$

Non-ℓ\ell-distance-balanced generalized Petersen graphs GP(n,3)GP(n,3) and GP(n,4)GP(n,4)

A connected graph $G$ of diameter ${\rm diam}(G) \ge \ell$ is $\ell$-distance-balanced if $|W_{xy}|=|W_{yx}|$ for every $x,y\in V(G)$ with $d_{G}(x,y)=\ell$, where $W_{xy}$ is the set of vertices of $G$ that are closer to $x$ than to $y$. We prove that the generalized Petersen graph $GP(n,3)$ where $n>16$ is not $\ell$-distance-balanced for any $1\le \ell < {\rm diam}(GP(n,3))$, and $GP(n,4)$ where $n>24$ is not $\ell$-distance-balanced for any $1\le \ell < {\rm diam}(GP(n,4))$. This partially solves a conjecture posed by \v{S}. Miklavi\v{c} and P. \v{S}parl (Discrete Appl. Math. 244:143-154, 2018).Comment: 3