Analytical study of degree-based graphical invariants in subdivided hexagonal cage networks

Muhammad Abdullah; Muhammad Usman Ghani; Hanen Karamti; Syed Ajaz K. Kirmani; Muhammad Abdullah; Muhammad Usman Ghani; Hanen Karamti; Syed Ajaz K. Kirmani

doi:10.3934/math.2026895

AIMS Mathematics

2026, Volume 11, Issue 7: 22114-22132. doi: 10.3934/math.2026895

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Research article

Analytical study of degree-based graphical invariants in subdivided hexagonal cage networks

1.
Department of Mathematics, Government College University, Lahore, Pakistan
2.
Institute of Mathematics, Khawaja Fareed University of Engineering & Information Technology, Abu Dhabi Road, 64200, Rahim Yar Khan, Pakistan
3.
Department of Computer Sciences, College of Computer and Information Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
4.
Department of Electrical Engineering, College of Engineering, Qassim University, Buraydah, Saudi Arabia

Received: 20 May 2026 Revised: 03 July 2026 Accepted: 10 July 2026 Published: 24 July 2026
MSC : 05C09, 05C12, 05C90

Graphical invariants and degree-based topological indices are important tools in graph theory, network science, and mathematical modeling for characterizing the structural properties of complex graph families. In this paper, we investigated subdivided hexagonal cage networks and derived exact analytical expressions for several important graphical invariants, including the general Randić index, the first Zagreb index, the second Zagreb index, the third Zagreb index, the atom-bond connectivity index, and the hyper-Zagreb index. By employing graph-theoretical techniques together with edge partition methods, explicit closed-form formulas were obtained in terms of the subdivision parameter of the underlying network. The derived results demonstrate the impact of subdivision operations on the structural behavior, branching characteristics, and quantitative complexity of the considered networks. Moreover, these graphical invariants provide useful insights into the structural characteristics, connectivity patterns, and complexity of the considered subdivision networks. The presented framework enriches the study of graphical invariants for nanostructures and complex graph and network models, while also contributing to applications in mathematical chemistry, materials science, and graph-based network analysis.
- graphical invariants,
- chemical graph theory,
- topological indices,
- degree-based indices,
- general Randić index,
- Zagreb indices,
- atom–bond connectivity index,
- hyper-Zagreb index,
- subdivided hexagonal cage networks
Citation: Muhammad Abdullah, Muhammad Usman Ghani, Hanen Karamti, Syed Ajaz K. Kirmani. Analytical study of degree-based graphical invariants in subdivided hexagonal cage networks[J]. AIMS Mathematics, 2026, 11(7): 22114-22132. doi: 10.3934/math.2026895

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Abstract

Graphical invariants and degree-based topological indices are important tools in graph theory, network science, and mathematical modeling for characterizing the structural properties of complex graph families. In this paper, we investigated subdivided hexagonal cage networks and derived exact analytical expressions for several important graphical invariants, including the general Randić index, the first Zagreb index, the second Zagreb index, the third Zagreb index, the atom-bond connectivity index, and the hyper-Zagreb index. By employing graph-theoretical techniques together with edge partition methods, explicit closed-form formulas were obtained in terms of the subdivision parameter of the underlying network. The derived results demonstrate the impact of subdivision operations on the structural behavior, branching characteristics, and quantitative complexity of the considered networks. Moreover, these graphical invariants provide useful insights into the structural characteristics, connectivity patterns, and complexity of the considered subdivision networks. The presented framework enriches the study of graphical invariants for nanostructures and complex graph and network models, while also contributing to applications in mathematical chemistry, materials science, and graph-based network analysis.

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