Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1878_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma =(K_n,H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a signed complete graph whose negative edges induce a subgraph H. Let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1878_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$A(\Gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the adjacency matrix of the signed graph <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1878_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. The largest eigenvalue of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1878_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$A(\Gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is called the index of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1878_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. In this paper, the index of all the signed complete graphs whose negative edges induce a bicyclic graph B is investigated. Specifically, the structure of the bicyclic graph B such that <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1878_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="89" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma =(K_n,B)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has the maximum index is determined.

The Maximum Index of Signed Complete Graphs

期刊论文
发表日期：

2025年5月20日

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Ziyi Fang,
Fan Chen,
Xiying Yuan

摘要

Let $\Gamma =(K_n,H)$ $Γ = (K_{n}, H)$ be a signed complete graph whose negative edges induce a subgraph H. Let $A(\Gamma )$ $A (Γ)$ be the adjacency matrix of the signed graph $\Gamma $ $Γ$ . The largest eigenvalue of $A(\Gamma )$ $A (Γ)$ is called the index of $\Gamma $ $Γ$ . In this paper, the index of all the signed complete graphs whose negative edges induce a bicyclic graph B is investigated. Specifically, the structure of the bicyclic graph B such that $\Gamma =(K_n,B)$ $Γ = (K_{n}, B)$ has the maximum index is determined.