<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\gamma _2(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\gamma _3(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the second and the third largest roots of the Laplacian matching polynomial of a simple graph <i>G</i>. In this note we structurally characterize the graphs in the sets <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {S}' \subset \mathcal {S}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">S</mi> </mrow> <mo>′</mo> </msup> <mo>⊂</mo> <mi mathvariant="script">S</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {S}= \{ G \mid \gamma _3(G) <2 \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">S</mi> <mo>=</mo> <mo stretchy="false">{</mo> <mi>G</mi> <mo>∣</mo> <msub> <mi>γ</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {S}'= \{ G \mid \gamma _2(G) \geqslant 2, \; \gamma _3(G) <2 \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">S</mi> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mi>G</mi> <mo>∣</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>⩾</mo> <mn>2</mn> <mo>,</mo> <mspace width="0.277778em" /> <msub> <mi>γ</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

On graphs whose third Laplacian matching root is small

期刊论文
出版模式：

开放获取
发表日期：

2025年12月21日

查看全文

Mengge Li,
Maurizio Brunetti,
Jianfeng Wang

摘要

Let $\gamma _2(G)$ $γ_{2} (G)$ and $\gamma _3(G)$ $γ_{3} (G)$ be the second and the third largest roots of the Laplacian matching polynomial of a simple graph G. In this note we structurally characterize the graphs in the sets $\mathcal {S}' \subset \mathcal {S}$ $S^{'} \subset S$ , where $\mathcal {S}= \{ G \mid \gamma _3(G) <2 \}$ $S = {G ∣ γ_{3} (G) < 2}$ and $\mathcal {S}'= \{ G \mid \gamma _2(G) \geqslant 2, \; \gamma _3(G) <2 \}$ $S^{'} = {G ∣ γ_{2} (G) ⩾ 2, γ_{3} (G) < 2}$ .