# Non-isomorphic Signatures on Some Generalised Petersen Graph

## TL;DR

This paper investigates the enumeration of non-isomorphic signatures on certain generalized Petersen graphs, providing counts for specific cases and proposing a conjecture for all odd-indexed graphs.

## Contribution

It offers the first known counts of signatures up to switching isomorphism for specific Petersen graphs and introduces a conjecture on the maximum size of signatures for odd-indexed graphs.

## Key findings

- Counted non-isomorphic signatures for P(3,1), P(5,1), P(7,1)
- Counted signatures of size two for P(2n+1,1) for all n ≥ 1
- Conjecture that signatures of P(2n+1,1) are at most size n+1

## Abstract

In this paper we find the number of different signatures of $P(3,1), P(5,1)$ and $P(7,1)$ upto switching isomorphism, where $P(n, k)$ denotes the generalised Petersen graph, $2k < n$. We also count the number of non-isomorphic signatures on $P(2n+1,1)$ of size two for all $n \geq 1$, and we conjecture that any signature of $P(2n+1,1)$, upto switching, is of size at most $n+1$.