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<p>The <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(A_{\alpha }\)</EquationSource> </InlineEquation> matrix of a graph <i>G</i> is defined as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(A_{\alpha }(G) = \alpha D(G) + (1-\alpha )A(G)\)</EquationSource> </InlineEquation>, where <i>D</i>(<i>G</i>) and <i>A</i>(<i>G</i>) denote the degree diagonal matrix and adjacency matrix of the graph <i>G</i>, respectively. In this article, we determine the eigenvalues of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(A_{\alpha }\)</EquationSource> </InlineEquation> matrix for the power graph of a class of metacyclic groups. We set upper and lower bounds for the largest eigenvalues of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(A_{\alpha }\)</EquationSource> </InlineEquation> matrix associated with the power graphs of the finite cyclic group of order <i>n</i> and the metacyclic group.</p>

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On the \(A_{\alpha }\) matrix over finite metacyclic groups

  • 期刊论文

  • 发表日期：

    2025年12月12日

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  • Aditya Singh,
  • Yogendra Singh,
  • Anand Kumar Tiwari,
  • S. Pirzada

摘要

The \(A_{\alpha }\) matrix of a graph G is defined as \(A_{\alpha }(G) = \alpha D(G) + (1-\alpha )A(G)\) , where D(G) and A(G) denote the degree diagonal matrix and adjacency matrix of the graph G, respectively. In this article, we determine the eigenvalues of \(A_{\alpha }\) matrix for the power graph of a class of metacyclic groups. We set upper and lower bounds for the largest eigenvalues of \(A_{\alpha }\) matrix associated with the power graphs of the finite cyclic group of order n and the metacyclic group.