Enumeration of dicirculant digraphs

TL;DR

This paper counts the number of dicirculant digraphs over the dicyclic group of order 4p, using Pólya's Enumeration Theorem, including connected graphs and those with specified out-degree.

Contribution

It provides a complete enumeration of dicirculant digraphs of order 4p, including connected cases and fixed out-degree, which was not previously known.

Findings

Number of dicirculant digraphs up to isomorphism calculated

Enumeration includes connected and out-degree specific graphs

Utilizes Pólya Enumeration Theorem for counting

Abstract

Let $T_{4p}=\langle a,b\mid a^{2p}=1,a^p=b^2, b^{-1}ab=a^{-1}\rangle$ be the dicyclic group of order $4p$. A Cayley digraph over $T_{4p}$ is called a dicirculant digraph. In this paper, we calculate the number of (connected) dicirculant digraphs of order $4p$ ($p$ prime) up to isomorphism by using the P\'olya Enumeration Theorem. Moreover, we get the number of (connected) dicirculant digraphs of order $4p$ ($p$ prime) and out-degree $k$ for every $k$.