Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$T_{4p}=\langle a,b\mid a^{2p}=1,a^p=b^2, b^{-1}ab=a^{-1}\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mrow> <mn>4</mn> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo stretchy="false">⟨</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∣</mo> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>a</mi> <mi>p</mi> </msup> <mo>=</mo> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>,</mo> <msup> <mi>b</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>a</mi> <mi>b</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">⟩</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the dicyclic group of order 4p. A Cayley digraph over <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$T_{4p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mrow> <mn>4</mn> <mi>p</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> 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.

Enumeration of dicirculant digraphs

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发表日期：

2026年5月2日

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Jing Wang,
Ligong Wang,
Xiaogang Liu

摘要

Let $T_{4p}=\langle a,b\mid a^{2p}=1,a^p=b^2, b^{-1}ab=a^{-1}\rangle $ $T_{4 p} = ⟨ a, b ∣ a^{2 p} = 1, a^{p} = b^{2}, b^{- 1} a b = a^{- 1} ⟩$ be the dicyclic group of order 4p. A Cayley digraph over $T_{4p}$ $T_{4 p}$ 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.