Proof of a conjecture on the geometric-quadratic index of unicyclic graphs and its applications

Abeer M. Albalahi; Kinkar Chandra Das; Sultan Ahmad; Tariq Alraqad; Akbar Ali; Abeer M. Albalahi; Kinkar Chandra Das; Sultan Ahmad; Tariq Alraqad; Akbar Ali

doi:10.3934/math.2026713

AIMS Mathematics

2026, Volume 11, Issue 6: 17437-17464. doi: 10.3934/math.2026713

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Proof of a conjecture on the geometric-quadratic index of unicyclic graphs and its applications

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 81451, Saudi Arabia
2.
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
3.
Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan

Received: 03 April 2026 Revised: 01 June 2026 Accepted: 11 June 2026 Published: 16 June 2026
MSC : 05C07, 05C09, 05C92

The geometric-quadratic (GQ) index is defined for a graph $ \Gamma $ as $ GQ(\Gamma) = \sum_{\nu_i \nu_j \in \mathcal{E}(\Gamma)} \sqrt{\frac{2\, d_i\, d_j}{d_i^2 + d_j^2}}, $ where $ d_i $ denotes the degree of the vertex $ \nu_i $. This degree-based topological index captures structural information by combining both geometric and quadratic contributions of adjacent vertex degrees. Recently, Furtula and Oz [Geometric-quadratic index from a mathematical perspective, Iranian J. Math. Chem., 16 (2025), 85-89] proposed a conjecture concerning the behavior of the GQ index for unicyclic graphs. In the present paper, we rigorously established the validity of this conjecture, thereby contributing to the theoretical understanding of degree-based graph invariants. Furthermore, Kumar and Das [Comparative study of GQ and QG indices as potentially favorable molecular descriptors, Int. J. Quantum Chem., 124 (2024), #27334] suggested that the GQ index may serve as a more effective molecular descriptor in quantitative structure-property relationship (QSPR) analysis, particularly for predicting physicochemical properties of molecular compounds beyond the extensively studied class of alkane isomers. Motivated by these findings, we further investigated the applicability of the GQ index by examining its role in elucidating QSPR in benzene-based hydrocarbons. For octane isomers, we extended this analysis using linear, quadratic, and cubic regression models across sixteen properties. The cubic model proved most effective for several properties, including four that previously failed under linear models alone. This analysis highlights the broader potential of the GQ index as a chemically meaningful descriptor, including its possible relevance in therapeutic and pharmaceutical contexts.
- molecular descriptor,
- geometric-quadratic (GQ) index,
- unicyclic graph,
- QSPR modeling,
- chemical graph theory
Citation: Abeer M. Albalahi, Kinkar Chandra Das, Sultan Ahmad, Tariq Alraqad, Akbar Ali. Proof of a conjecture on the geometric-quadratic index of unicyclic graphs and its applications[J]. AIMS Mathematics, 2026, 11(6): 17437-17464. doi: 10.3934/math.2026713

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Abstract

The geometric-quadratic (GQ) index is defined for a graph $ \Gamma $ as $ GQ(\Gamma) = \sum_{\nu_i \nu_j \in \mathcal{E}(\Gamma)} \sqrt{\frac{2\, d_i\, d_j}{d_i^2 + d_j^2}}, $ where $ d_i $ denotes the degree of the vertex $ \nu_i $. This degree-based topological index captures structural information by combining both geometric and quadratic contributions of adjacent vertex degrees. Recently, Furtula and Oz [Geometric-quadratic index from a mathematical perspective, Iranian J. Math. Chem., 16 (2025), 85-89] proposed a conjecture concerning the behavior of the GQ index for unicyclic graphs. In the present paper, we rigorously established the validity of this conjecture, thereby contributing to the theoretical understanding of degree-based graph invariants. Furthermore, Kumar and Das [Comparative study of GQ and QG indices as potentially favorable molecular descriptors, Int. J. Quantum Chem., 124 (2024), #27334] suggested that the GQ index may serve as a more effective molecular descriptor in quantitative structure-property relationship (QSPR) analysis, particularly for predicting physicochemical properties of molecular compounds beyond the extensively studied class of alkane isomers. Motivated by these findings, we further investigated the applicability of the GQ index by examining its role in elucidating QSPR in benzene-based hydrocarbons. For octane isomers, we extended this analysis using linear, quadratic, and cubic regression models across sixteen properties. The cubic model proved most effective for several properties, including four that previously failed under linear models alone. This analysis highlights the broader potential of the GQ index as a chemically meaningful descriptor, including its possible relevance in therapeutic and pharmaceutical contexts.

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