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On Unicyclic Graphs with a Given Number of Pendent Vertices or Matching Number and Their Graphical Edge-Weight-Function Indices

Author

Listed:
  • Akbar Ali

    (Department of Mathematics, College of Science, University of Ha’il, Ha’il P.O. Box 2440, Saudi Arabia)

  • Abdulaziz M. Alanazi

    (Department of Mathematics, Faculty of Sciences, University of Tabuk, Tabuk P.O. Box 741, Saudi Arabia)

  • Taher S. Hassan

    (Department of Mathematics, College of Science, University of Ha’il, Ha’il P.O. Box 2440, Saudi Arabia
    Mathematics Section, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy
    Department of Mathematics, Mansoura University, Mansoura 35516, Egypt)

  • Yilun Shang

    (Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK)

Abstract

Consider a unicyclic graph G with edge set E ( G ) . Let f be a real-valued symmetric function defined on the Cartesian square of the set of all distinct elements of G ’s degree sequence. A graphical edge-weight-function index of G is defined as I f ( G ) = ∑ x y ∈ E ( G ) f ( d G ( x ) , d G ( y ) ) , where d G ( x ) denotes the degree a vertex x in G . This paper determines optimal bounds for I f ( G ) in terms of the order of G and a parameter z , where z is either the number of pendent vertices of G or the matching number of G . The paper also fully characterizes all unicyclic graphs that achieve these bounds. The function f must satisfy specific requirements, which are met by several popular indices, including the Sombor index (and its reduced version), arithmetic–geometric index, sigma index, and symmetric division degree index. Consequently, the general results obtained provide bounds for several well-known indices.

Suggested Citation

  • Akbar Ali & Abdulaziz M. Alanazi & Taher S. Hassan & Yilun Shang, 2024. "On Unicyclic Graphs with a Given Number of Pendent Vertices or Matching Number and Their Graphical Edge-Weight-Function Indices," Mathematics, MDPI, vol. 12(23), pages 1-12, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:23:p:3658-:d:1527174
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    References listed on IDEAS

    as
    1. Shang, Yilun, 2022. "Sombor index and degree-related properties of simplicial networks," Applied Mathematics and Computation, Elsevier, vol. 419(C).
    2. Zheng, Ruiling & Su, Peifeng & Jin, Xian’an, 2023. "Arithmetic-geometric matrix of graphs and its applications," Applied Mathematics and Computation, Elsevier, vol. 442(C).
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