Diameter of generalized Petersen graphs

TL;DR

This paper investigates the diameter of generalized Petersen graphs, providing exact formulas, conditions for specific diameter values, and an efficient algorithm for computation.

Contribution

It offers necessary and sufficient conditions for the diameter of GPG(n,s) to be d+1 or d+2, along with exact diameter values for most cases and a logarithmic time algorithm.

Findings

Conditions for diameter d+1 and d+2 are established.

Exact diameters are computed for most cases.

An O(log n) algorithm for diameter calculation is presented.

Abstract

Due to their broad application to different fields of theory and practice, generalized Petersen graphs $GPG(n,s)$ have been extensively investigated. Despite the regularity of generalized Petersen graphs, determining an exact formula for the diameter is still a difficult problem. In their paper, Beenker and Van Lint have proved that if the circulant graph $C_n(1,s)$ has diameter $d$, then $GPG(n,s)$ has diameter at least $d+1$ and at most $d+2$. In this paper, we provide necessary and sufficient conditions so that the diameter of $GPG(n,s)$ is equal to $d+1,$ and sufficient conditions so that the diameter of $GPG(n,s)$ is equal to $d+2.$ Afterwards, we give exact values for the diameter of $GPG(n,s)$ for almost all cases of $n$ and $s.$ Furthermore, we show that there exists an algorithm computing the diameter of generalized Petersen graphs with running time $O$(log$n$).