Given a connected graph G with adjacency matrix A(G) and distance matrix D(G), its generalized adjacency-distance matrix is defined as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>α</mi> <mi>D</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>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq2"> <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 refer to the largest eigenvalue of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$S_\alpha (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> as the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$S_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>-spectral radius of G. This paper presents some local graft transformations which change the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$S_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>-spectral radius of graphs. Using these transformations, we characterize the extremal graphs with maximum <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$S_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>-spectral radius in all n-vertex connected graphs with given clique number or maximum degree. Additionally, we also characterize the extremal graph with minimum <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$S_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>-spectral radius among the most of bicyclic graphs with n vertices.

Further results on the generalized adjacency-distance spectral radius of graphs

期刊论文
发表日期：

2025年7月29日

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Jia-Long Wu,
Gui-Xian Tian,
Shu-Yu Cui,
Jia-Hui Li

摘要

Given a connected graph G with adjacency matrix A(G) and distance matrix D(G), its generalized adjacency-distance matrix is defined as $S_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$ $S_{α} (G) = α D (G) + (1 - α) A (G)$ for $\alpha \in [0,1]$ $α \in [0, 1]$ . We refer to the largest eigenvalue of $S_\alpha (G)$ $S_{α} (G)$ as the $S_\alpha $ $S_{α}$ -spectral radius of G. This paper presents some local graft transformations which change the $S_\alpha $ $S_{α}$ -spectral radius of graphs. Using these transformations, we characterize the extremal graphs with maximum $S_\alpha $ $S_{α}$ -spectral radius in all n-vertex connected graphs with given clique number or maximum degree. Additionally, we also characterize the extremal graph with minimum $S_\alpha $ $S_{α}$ -spectral radius among the most of bicyclic graphs with n vertices.