Self-centeredness of Generalized Petersen Graphs

Abstract

A connected graph is said to be self-centered if all its vertices have the same eccentricity. The family of generalized Petersen graphs P(n, k), introduced by Coxeter [6] and named by Watkins [18], is a family of cubic graphs of order 2n defined by positive integral parameters n and k, $n\ge 2k$. Not all generalized Petersen graphs are self-centered. In this paper, we prove self-centeredness of P(n, k) whenever k divides n and $k< \frac{n}{2}$, except the case when n is odd and k is even. We also prove non-self-centeredness of generalized Petersen graphs P(n, k) when n even with $k=\frac{n}{2}$; $n=4m+2$ with $k=\frac{n}{2}-1$ for some positive integer $m\ge 3$; $n\ge 9$ is odd and $k=2$ or $k=\frac{n-1}{2}$; and $n = m(4m+1 )\pm (m+1)$ with $k = 4m + 1$ for any positive integer $m \ge 2$. Finally, we make an exhaustive computer search and get all possible values of n and k for which P(n, k) is non-self-centered.