Let G be a connected graph and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha \in [0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a real number. The <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ D_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>-matrix of G is defined as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$D_\alpha (G)=\alpha Tr(G)+(1-\alpha )D(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>α</mi> <mi>T</mi> <mi>r</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mi>D</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where Tr(G) is the diagonal matrix of vertex transmissions of G and D(G) is the distance matrix of G. The largest eigenvalue of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$D_\alpha (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is called the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ D_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>-spectral radius of G. For <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha \in [0,\frac{1}{2}]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, we determine the unique graph with the minimum <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$D_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>-spectral radius among n-vertex graphs with given matching number <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$m\le \frac{n}{4}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≤</mo> <mfrac> <mi>n</mi> <mn>4</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, and the unique graph with the minimum <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$D_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>-spectral radius among n-vertex graphs with <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> pendent vertices. For <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\alpha \in [0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we determine the unique graph with the minimum <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$D_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>-spectral radius among n-vertex graphs with <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$0 \le r \le n-3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>r</mi> <mo>≤</mo> <mi>n</mi> <mo>-</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> pendent vertices and the unique graph with the minimum <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$D_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>-spectral radius among n-vertex trees with given maximum degree <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$ \Delta \ge \lceil \frac{n}{2}\rceil $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mo>≥</mo> <mo>⌈</mo> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> <mo>⌉</mo> </mrow> </math></EquationSource> </InlineEquation>.

Extremal Graphs with Respect to the $D_\alpha $-spectral Radius

期刊论文
发表日期：

2026年6月2日

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Xiaohan Liu,
Aimei Yu,
Rong-Xia Hao

摘要

Let G be a connected graph and $\alpha \in [0,1)$ $α \in [0, 1)$ be a real number. The $ D_\alpha $ $D_{α}$ -matrix of G is defined as $D_\alpha (G)=\alpha Tr(G)+(1-\alpha )D(G)$ $D_{α} (G) = α T r (G) + (1 - α) D (G)$ , where Tr(G) is the diagonal matrix of vertex transmissions of G and D(G) is the distance matrix of G. The largest eigenvalue of $D_\alpha (G)$ $D_{α} (G)$ is called the $ D_\alpha $ $D_{α}$ -spectral radius of G. For $\alpha \in [0,\frac{1}{2}]$ $α \in [0, \frac{1}{2}]$ , we determine the unique graph with the minimum $D_\alpha $ $D_{α}$ -spectral radius among n-vertex graphs with given matching number $m\le \frac{n}{4}$ $m \leq \frac{n}{4}$ , and the unique graph with the minimum $D_\alpha $ $D_{α}$ -spectral radius among n-vertex graphs with $n-2$ $n - 2$ pendent vertices. For $\alpha \in [0,1)$ $α \in [0, 1)$ , we determine the unique graph with the minimum $D_\alpha $ $D_{α}$ -spectral radius among n-vertex graphs with $0 \le r \le n-3$ $0 \leq r \leq n - 3$ pendent vertices and the unique graph with the minimum $D_\alpha $ $D_{α}$ -spectral radius among n-vertex trees with given maximum degree $ \Delta \ge \lceil \frac{n}{2}\rceil $ $Δ \geq ⌈ \frac{n}{2} ⌉$ .

Extremal Graphs with Respect to the \(D_\alpha \)-spectral Radius

摘要