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<p>Among the set of outerplanar graphs on <i>n</i> vertices with a forbidden structure <i>F</i>, we obtain that the extremal graph having the maximum spectral radius contains a subgraph <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(K_{1,n-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>, where <i>F</i> is an arbitrary graph in a large family of outerplanar graphs with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n \geqslant 400\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>400</mn> </mrow> </math></EquationSource> </InlineEquation>. Based on our new results obtained here, we characterize the unique graph having the maximum spectral radius among the set of outerplanar graphs on <i>n</i> vertices with a forbidden structure <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(F_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(F_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> is a friendship graph, a given number of independent edges, a fan graph, a cycle, and a generalized theta graph.</p>

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Extremal Spectral Radius of Outplanar Graphs with Forbidden Structures

  • 期刊论文

  • 发表日期：

    2025年6月28日

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  • Wen-Huan Wang,
  • Huan-Lei Shi,
  • Yuan-Zheng-Fang Sun

摘要

Among the set of outerplanar graphs on n vertices with a forbidden structure F, we obtain that the extremal graph having the maximum spectral radius contains a subgraph \(K_{1,n-1}\) K 1 , n - 1 , where F is an arbitrary graph in a large family of outerplanar graphs with \(n \geqslant 400\) n ⩾ 400 . Based on our new results obtained here, we characterize the unique graph having the maximum spectral radius among the set of outerplanar graphs on n vertices with a forbidden structure \(F_1\) F 1 , where \(F_1\) F 1 is a friendship graph, a given number of independent edges, a fan graph, a cycle, and a generalized theta graph.