<p>The cyclic subgroup graph <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1735_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{\Gamma }(\mathscr {G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold">Γ</mi> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of a group <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1735_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathscr {G}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation> is the graph with vertices are cyclic subgroups of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1735_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathscr {G}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation> and two distinct vertices <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1735_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">H</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1735_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">H</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> are adjacent if and only if <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1735_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}_1 \le \mathcal {H}_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">H</mi> <mn>1</mn> </msub> <mo>≤</mo> <msub> <mi mathvariant="script">H</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> and there does not exist any cyclic subgroup <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1735_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">K</mi> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1735_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}_1< \mathcal {K} < \mathcal {H}_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">H</mi> <mn>1</mn> </msub> <mo><</mo> <mi mathvariant="script">K</mi> <mo><</mo> <msub> <mi mathvariant="script">H</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we classify all the finite groups <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1735_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathscr {G}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1735_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{\Gamma }(\mathscr {G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold">Γ</mi> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is the line graph of some graph.</p>

Line Graph Characterization of Cyclic Subgroup Graph

期刊论文
发表日期：

2025年7月26日

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Siddharth Malviy,
Vipul Kakkar,
Swapnil Srivastava

摘要

The cyclic subgroup graph $\varvec{\Gamma }(\mathscr {G})$ $Γ (G)$ of a group $\mathscr {G}$ $G$ is the graph with vertices are cyclic subgroups of $\mathscr {G}$ $G$ and two distinct vertices $\mathcal {H}_1$ $H_{1}$ and $\mathcal {H}_2$ $H_{2}$ are adjacent if and only if $\mathcal {H}_1 \le \mathcal {H}_2$ $H_{1} \leq H_{2}$ and there does not exist any cyclic subgroup $\mathcal {K}$ $K$ such that $\mathcal {H}_1< \mathcal {K} < \mathcal {H}_2$ $H_{1} < K < H_{2}$ . In this paper, we classify all the finite groups $\mathscr {G}$ $G$ such that $\varvec{\Gamma }(\mathscr {G})$ $Γ (G)$ is the line graph of some graph.