Enumeration of dicirculant digraphs

Abstract

Let $T_{4p}=⟨a,b\mid a^{2p}=1,a^p=b^2,b^{-1}ab=a^{-1}⟩$ 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 Pólya Enumeration Theorem. Moreover, we get the number of (connected) dicirculant digraphs of order $4p$ ($p$ prime) and out-degree $k$ for every $k$.