Abstract <p> Let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3028_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> be a finite simple graph on the vertex set <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3028_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(V\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3028_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(I_G\)</EquationSource> </InlineEquation> its edge ideal in the polynomial ring <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3028_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">\(S=\mathbb{K}[x_V]\)</EquationSource> </InlineEquation>. In this paper, we compute the Castelnuovo–Mumford regularity and depth of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3028_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(S/I_G\)</EquationSource> </InlineEquation> when <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3028_Article_IEq8.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="97" /> </InlineMediaObject> <EquationSource Format="TEX">\(G=F_{k}^{W}(K_n)\)</EquationSource> </InlineEquation> is a <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3028_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(k\)</EquationSource> </InlineEquation>-fan graph, or <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3028_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">\(G=G_1\circ G_2\)</EquationSource> </InlineEquation> or <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3028_Article_IEq11.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">\(G=G_1* G_2\)</EquationSource> </InlineEquation> is the graph obtained from fan graphs <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3028_Article_IEq12.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(G_1\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3028_Article_IEq13.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(G_2\)</EquationSource> </InlineEquation> by the <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3028_Article_IEq1.gif" Format="GIF" Height="9" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\circ\)</EquationSource> </InlineEquation> operation or the <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3028_Article_IEq2.gif" Format="GIF" Height="9" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(*\)</EquationSource> </InlineEquation> operation, respectively. </p>

The \(\circ\) Operation and \(*\) Operation of Fan Graphs

期刊论文
发表日期：

2025年2月1日

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G. Zhu,
Y. Cui,
Y. Yang,
Y. Yang

摘要

Abstract

Let \(G\) be a finite simple graph on the vertex set \(V\) and \(I_G\) its edge ideal in the polynomial ring \(S=\mathbb{K}[x_V]\) . In this paper, we compute the Castelnuovo–Mumford regularity and depth of \(S/I_G\) when \(G=F_{k}^{W}(K_n)\) is a \(k\) -fan graph, or \(G=G_1\circ G_2\) or \(G=G_1* G_2\) is the graph obtained from fan graphs \(G_1\) , \(G_2\) by the \(\circ\) operation or the \(*\) operation, respectively.