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Given two signed graphs \(\varGamma _1\) with node set \(\{u_1,u_2,\ldots ,u_n\}\) and \(\varGamma _2\) , the neighbourhood corona, \(\varGamma _1*\varGamma _2\) is the signed graph obtained by taking one copy of \(\varGamma _1\) and \(n_1\) copies of \(\varGamma _2\) , and joining every neighbour of the ith node with each node of the ith copy of \(\varGamma _2\) by a new signed edge. This paper will determine the condition for \(\varGamma _1*\varGamma _2\) to be balanced. We also determine the adjacency spectrum of \(\varGamma _1*\varGamma _2\) for arbitrary \(\varGamma _1\) and \(\varGamma _2\) , and Laplacian and signless Laplacian spectrum of \(\varGamma _1*\varGamma _2\) for regular \(\varGamma _1\) and arbitrary \(\varGamma _2\) , in terms of the corresponding spectrum of \(\varGamma _1\) and \(\varGamma _2\) .

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On Spectrum of Neighbourhood Corona Product of Signed Graphs

  • 会议论文

  • 发表日期：

    2025年

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  • Bishal Sonar,
  • Satyam Guragain,
  • Ravi Srivastava

摘要

Given two signed graphs \(\varGamma _1\) with node set \(\{u_1,u_2,\ldots ,u_n\}\) and \(\varGamma _2\) , the neighbourhood corona, \(\varGamma _1*\varGamma _2\) is the signed graph obtained by taking one copy of \(\varGamma _1\) and \(n_1\) copies of \(\varGamma _2\) , and joining every neighbour of the ith node with each node of the ith copy of \(\varGamma _2\) by a new signed edge. This paper will determine the condition for \(\varGamma _1*\varGamma _2\) to be balanced. We also determine the adjacency spectrum of \(\varGamma _1*\varGamma _2\) for arbitrary \(\varGamma _1\) and \(\varGamma _2\) , and Laplacian and signless Laplacian spectrum of \(\varGamma _1*\varGamma _2\) for regular \(\varGamma _1\) and arbitrary \(\varGamma _2\) , in terms of the corresponding spectrum of \(\varGamma _1\) and \(\varGamma _2\) .