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Normalized Laplacian Spectrum and Graph Invariant Formulas of Polygonized Graphs Based on Tridiagonal Matrices

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  • Hao Li
  • Xinyi Chen
  • Hao Liu

Abstract

The polygonized graph Pn,kG is constructed from a simple connected graph G through a substitution process. During this process, each edge in G is replaced by one path of length 1 and k paths of length +1n,k≥1. Based on the properties of the determinants of tridiagonal matrices, we present a unified formula for computing the normalized Laplacian spectrum of Pn,kG from that of G. Moreover, we offer explicit formulas for calculating the number of spanning trees, Kemeny’s constant, and the multiplicative degree−Kirchhoff index of Pn,kG. In the educational context of graph theory and linear algebra, determinants serve as a valuable tool for exploring relevant graph parameters.

Suggested Citation

  • Hao Li & Xinyi Chen & Hao Liu, 2026. "Normalized Laplacian Spectrum and Graph Invariant Formulas of Polygonized Graphs Based on Tridiagonal Matrices," Journal of Mathematics, Hindawi, vol. 2026, pages 1-12, April.
    • Handle: RePEc:hin:jjmath:4974252
    DOI: 10.1155/jom/4974252
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