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

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.