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Proof of an open problem on the maximization of the Euler–Sombor index in chemical unicyclic graphs

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  • Ahmad, Sultan
  • Das, Kinkar Chandra

Abstract

A topological index is a numerical property of a molecular graph that reflects its structural features. The geometric interpretation and the capacity of topological indices to distinguish between molecular structures have made them an important focus of current research. In this line, numerous degree-based indices have been introduced in recent years. Among these, the Euler–Sombor index, derived from Euler’s approximation formula for the perimeter of an ellipse, has attracted particular attention. For a graph Γ, the Euler–Sombor index (abbreviated as EU–index) is defined as: EU(Γ)=∑νiνj∈E(Γ)di2+dj2+didj,where E(Γ) denotes the edge set and di is the degree of a vertex νi in Γ. Quite recently, Khanra and Das (2025) posed a problem on characterizing chemical unicyclic graphs with respect to the EU–index in terms of graph order, addressing both the maximizing and minimizing cases. This problem was subsequently discussed by Das et al. (in press), where the minimizing case was completely resolved, while the maximizing case remained open. In this paper, we present a complete characterization of the maximizing problem and identify the corresponding extremal graphs.

Suggested Citation

  • Ahmad, Sultan & Das, Kinkar Chandra, 2026. "Proof of an open problem on the maximization of the Euler–Sombor index in chemical unicyclic graphs," Chaos, Solitons & Fractals, Elsevier, vol. 203(C).
    • Handle: RePEc:eee:chsofr:v:203:y:2026:i:c:s0960077925017333
    DOI: 10.1016/j.chaos.2025.117720
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    References listed on IDEAS

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    1. Das, Kinkar Chandra & Gutman, Ivan, 2022. "On Sombor index of trees," Applied Mathematics and Computation, Elsevier, vol. 412(C).
    2. Ahmad, Sultan & Das, Kinkar Chandra, 2025. "A complete solution for maximizing the general Sombor index of chemical trees with given number of pendant vertices," Applied Mathematics and Computation, Elsevier, vol. 505(C).
    3. Wu, Zhiqiang & Xue, Yumei & He, Huixia & Zeng, Cheng & Wang, Wenjie, 2024. "Kirchhoff index of Vicsek polygon networks and its applications," Chaos, Solitons & Fractals, Elsevier, vol. 184(C).
    4. Noureen, Sadia & Bhatti, Akhlaq Ahmad & Ali, Akbar, 2021. "Towards the solution of an extremal problem concerning the Wiener polarity index of alkanes," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    5. Raza, Zahid & Akhter, Shehnaz, 2023. "On maximum Zagreb connection indices for trees with fixed domination number," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    6. Kinkar Chandra Das & Yilun Shang, 2021. "Some Extremal Graphs with Respect to Sombor Index," Mathematics, MDPI, vol. 9(11), pages 1-15, May.
    7. Das, Kinkar Chandra, 2024. "Open problems on Sombor index of unicyclic and bicyclic graphs," Applied Mathematics and Computation, Elsevier, vol. 473(C).
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