<p>For <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ u_1, u_2 \in V(G) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>∈</mo> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the closeness matrix of a graph <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ G $</EquationSource> <EquationSource Format="MATHML"><math> <mi>G</mi> </math></EquationSource> </InlineEquation>, denoted by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ C(G) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, is a symmetric matrix where each entry is defined as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$c_G(u_1, u_2) = 2^{-d(u_1,u_2)}, \; \text {for } u_1 \ne u_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow> <mo>-</mo> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mspace width="0.277778em" /> <mtext>for</mtext> <mspace width="0.333333em" /> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>≠</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$c_G(u_1, u_2) = 0, \; \text {if } u_1 = u_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.277778em" /> <mtext>if</mtext> <mspace width="0.333333em" /> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ d(u_1, u_2) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> represents the distance between vertices <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ u_1 $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ u_2 $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ G $</EquationSource> <EquationSource Format="MATHML"><math> <mi>G</mi> </math></EquationSource> </InlineEquation>. Let the eigenvalues of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$ C(G) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be ordered as <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\rho _1(C(G)) \ge \rho _2(C(G)) \ge \cdots \ge \rho _n(C(G))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ρ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <msub> <mi>ρ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mo>⋯</mo> <mo>≥</mo> <msub> <mi>ρ</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we identify graphs for which the second-largest closeness eigenvalue satisfies <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$ \rho _2(C(G)) \le 0 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ρ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Specifically, we characterize such graphs within the families of trees, unicyclic graphs, and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$K_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>-minor-free multicyclic graphs.</p>

On graphs with second largest closeness eigenvalue at most zero

期刊论文
发表日期：

2026年3月3日

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Qaisar Farhad,
Fazal Hayat,
Shou-Jun Xu

摘要

For $ u_1, u_2 \in V(G) $ $u_{1}, u_{2} \in V (G)$ , the closeness matrix of a graph $ G $ $G$ , denoted by $ C(G) $ $C (G)$ , is a symmetric matrix where each entry is defined as $c_G(u_1, u_2) = 2^{-d(u_1,u_2)}, \; \text {for } u_1 \ne u_2$ $c_{G} (u_{1}, u_{2}) = 2^{- d (u_{1}, u_{2})}, for u_{1} \neq u_{2}$ , and $c_G(u_1, u_2) = 0, \; \text {if } u_1 = u_2$ $c_{G} (u_{1}, u_{2}) = 0, if u_{1} = u_{2}$ , where $ d(u_1, u_2) $ $d (u_{1}, u_{2})$ represents the distance between vertices $ u_1 $ $u_{1}$ and $ u_2 $ $u_{2}$ in $ G $ $G$ . Let the eigenvalues of $ C(G) $ $C (G)$ be ordered as $\rho _1(C(G)) \ge \rho _2(C(G)) \ge \cdots \ge \rho _n(C(G))$ $ρ_{1} (C (G)) \geq ρ_{2} (C (G)) \geq \dots \geq ρ_{n} (C (G))$ . In this paper, we identify graphs for which the second-largest closeness eigenvalue satisfies $ \rho _2(C(G)) \le 0 $ $ρ_{2} (C (G)) \leq 0$ . Specifically, we characterize such graphs within the families of trees, unicyclic graphs, and $K_4$ $K_{4}$ -minor-free multicyclic graphs.