Enumeration of Cayley graphs over a nonabelian group of order $8p$

TL;DR

This paper derives formulas to count all and connected Cayley graphs over a specific nonabelian group of order 8p, using Pólya Enumeration, and provides explicit counts for primes 3 to 13.

Contribution

It introduces explicit formulas for enumerating Cayley graphs over a nonabelian group of order 8p, including connected graphs, expanding combinatorial understanding.

Findings

Formulas for total Cayley graphs over T_{8p}

Formulas for connected Cayley graphs over T_{8p}

Exact counts for primes p from 3 to 13

Abstract

Let $T_{8p} = \left\langle a,b\mid a^{2p}=b^8=e,a^p=b^4,b^{-1}ab=a^{-1} \right\rangle$ be a nonabelian group of order $8p$, where $p$ is an odd prime number. In this paper, we give the formula to calculate the number of Cayley graphs over $T_{8p}$ up to isomorphism by using the P\'olya Enumeration Theorem. Moreover, we get the formula to calculate the number of connected Cayley graphs over $T_{8p}$ by deleting the disconnected graphs. By applying the results, we list the exact number of (connected) Cayley graphs for $3\leq p \leq 13$.