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Resolving an Open Problem on the Exponential Arithmetic–Geometric Index of Unicyclic Graphs

Author

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  • Kinkar Chandra Das

    (Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea)

  • Jayanta Bera

    (Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea)

Abstract

Recently, the exponential arithmetic–geometric index ( E A G ) was introduced. The exponential arithmetic–geometric index ( E A G ) of a graph G is defined as E A G ( G ) = ∑ v i v j ∈ E ( G ) e d i + d j 2 d i d j , where d i represents the degree of the vertex v i in G . The characterization of extreme structures in relation to graph invariants from the class of unicyclic graphs is an important problem in discrete mathematics. Cruz et al., 2022 proposed a unified method for finding extremal unicyclic graphs for exponential degree-based graph invariants. However, in the case of E A G , this method is insufficient for generating the maximal unicyclic graph. Consequently, the same article presented an open problem for the investigation of the maximal unicyclic graph with respect to this invariant. This article completely characterizes the maximal unicyclic graph in relation to E A G .

Suggested Citation

  • Kinkar Chandra Das & Jayanta Bera, 2025. "Resolving an Open Problem on the Exponential Arithmetic–Geometric Index of Unicyclic Graphs," Mathematics, MDPI, vol. 13(9), pages 1-11, April.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:9:p:1391-:d:1641665
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    References listed on IDEAS

    as
    1. Kinkar Chandra Das, 2025. "On the Exponential Atom-Bond Connectivity Index of Graphs," Mathematics, MDPI, vol. 13(2), pages 1-19, January.
    2. Yajing Wang & Yubin Gao, 2020. "Nordhaus–Gaddum-Type Relations for Arithmetic-Geometric Spectral Radius and Energy," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-7, July.
    3. Mondal, Sourav & Das, Kinkar Chandra, 2024. "Complete solution to open problems on exponential augmented Zagreb index of chemical trees," Applied Mathematics and Computation, Elsevier, vol. 482(C).
    4. Chen, Meng & Zhu, Yan, 2024. "Extremal unicyclic graphs of Sombor index," Applied Mathematics and Computation, Elsevier, vol. 463(C).
    5. Snježana Majstorović Ergotić, 2024. "On Unicyclic Graphs with Minimum Graovac–Ghorbani Index," Mathematics, MDPI, vol. 12(3), pages 1-17, January.
    6. V. S. Shigehalli & Rachanna Kanabur, 2016. "Computation of New Degree-Based Topological Indices of Graphene," Journal of Mathematics, Hindawi, vol. 2016, pages 1-6, September.
    7. Roberto Cruz & Juan Daniel Monsalve & Juan Rada, 2021. "The balanced double star has maximum exponential second Zagreb index," Journal of Combinatorial Optimization, Springer, vol. 41(2), pages 544-552, February.
    8. Akbar Jahanbani & Murat Cancan & Ruhollah Motamedi & M. T. Rahim, 2022. "Extremal Trees for the Exponential of Forgotten Topological Index," Journal of Mathematics, Hindawi, vol. 2022, pages 1-8, February.
    9. 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.
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