# On Cayley representations of finite graphs over abelian p-groups

## TL;DR

This paper presents a polynomial-time algorithm to find all non-equivalent Cayley representations of a graph over specific abelian p-groups, enabling efficient recognition and isomorphism testing of such Cayley graphs.

## Contribution

It introduces a novel polynomial-time algorithm for identifying all Cayley representations over groups of the form C_p × C_{p^k} for p=2,3, advancing graph isomorphism research.

## Key findings

- Algorithm finds all non-equivalent Cayley representations efficiently.
- Recognition and isomorphism problems for these Cayley graphs are solvable in polynomial time.
- Provides a constructive method for Cayley graph analysis over abelian p-groups.

## Abstract

We construct a polynomial-time algorithm which given a graph $\Gamma$ finds the full set of non-equivalent Cayley representations of $\Gamma$ over the group $D\cong C_p\times C_{p^k}$, where $p\in\{2,3\}$ and $k\geq 1$. This result implies that the recognition and the isomorphism problems for Cayley graphs over $D$ can be solved in polynomial time.