# On the Enumeration of Circulant Graphs of Prime-Power Order: the case of $p^3$

## TL;DR

This paper extends existing methods to enumerate non-isomorphic circulant graphs of order p^3, specifically for p=3 and p=5, using computational tools to address a longstanding combinatorial problem.

## Contribution

It develops the application of multiplier and structural methods to count circulant graphs of order p^3, advancing enumeration techniques for prime-power orders.

## Key findings

- Successfully enumerated circulant graphs of order 27 and 125.
- Extended enumeration methods to prime-power order p^3.
- Utilized GAP software for computational enumeration.

## Abstract

A well-known problem in Algebraic Combinatorics, is the enumeration of circulant graphs. The failure of Adam's Conjecture for such graphs with order containing a repeated prime, led researchers to investigate the problem using two different methods, namely the multiplier method and the structural method. The former makes use of isomorphism theorems whereas the latter involves Schur rings. Both these methods have already been used to count the number of non-isomorphic circulants of order $p^2$. This research focuses on the extension of these two methods to enumerate circulants of order $p^3$, in particular for $p=3$ and $p=5$, through the use of the computer package GAP.