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Abstract <p> Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> be a connected graph of order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation>. For any integer <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(k\geq2\)</EquationSource> </InlineEquation>, a spanning <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(k\)</EquationSource> </InlineEquation>-tree of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> is a spanning tree in which every vertex has degree at most <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(k\)</EquationSource> </InlineEquation>. In this paper, we provide a tight <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(A_{\alpha}\)</EquationSource> </InlineEquation>-spectral condition to guarantee the existence of a spanning <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(k\)</EquationSource> </InlineEquation>-tree in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> with extremal graphs being characterized. Moreover, we also present tight <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(A_{\alpha}\)</EquationSource> </InlineEquation>-spectral conditions for <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> admitting a spanning <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(k\)</EquationSource> </InlineEquation>-ended-tree (i.e., a spanning tree with at most <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(k\)</EquationSource> </InlineEquation> leaves) and determine the extremal graphs. </p>

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The \(A_{\alpha}\)-Spectral Radius and Spanning Trees of Graphs

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • J. Ha,
  • F. Wen,
  • Y. Chen

摘要

Abstract

Let \(G\) be a connected graph of order \(n\) . For any integer \(k\geq2\) , a spanning \(k\) -tree of \(G\) is a spanning tree in which every vertex has degree at most \(k\) . In this paper, we provide a tight \(A_{\alpha}\) -spectral condition to guarantee the existence of a spanning \(k\) -tree in \(G\) with extremal graphs being characterized. Moreover, we also present tight \(A_{\alpha}\) -spectral conditions for \(G\) admitting a spanning \(k\) -ended-tree (i.e., a spanning tree with at most \(k\) leaves) and determine the extremal graphs.