On {1,2}-distance-balancedness of generalized Petersen graphs

Abstract

A connected graph G of diameter $\mathrm{diam}\left(G\right)\ge \ell$ is ℓ-distance-balanced if $\left|W_{xy}\right|=\left|W_{yx}\right|$ for every $x,y\in V\left(G\right)$ with $d_G(x,y)=\ell$, where $W_{xy}$ is the set of vertices of G that are closer to x than to y. It is proved that if $k\ge 3$ and $n>k(k+2)$, then the generalized Petersen graph $GP(n,k)$ is not distance-balanced and that $GP\left(k\right(k+2),k)$ is distance-balanced. It is also proved that if $k\ge 6$ where k is even, and $n>\frac{1}{2}{k}^3-\frac{3}{4}{k}^2+9k$, or if $k\ge 5$ where k is odd, and $n>\frac{1}{2}{k}^3-\frac{1}{4}{k}^2+\frac{31}{4}k$, then $GP(n,k)$ is not 2-distance-balanced.