The elliptic Dharwad index for structural characterizations of extremal graphs and QSPR applications

Suha Wazzan; Mohammed Alsharafi; Suha Wazzan; Mohammed Alsharafi

doi:10.3934/math.2026999

AIMS Mathematics

2026, Volume 11, Issue 8: 24839-24876. doi: 10.3934/math.2026999

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The elliptic Dharwad index for structural characterizations of extremal graphs and QSPR applications

Suha Wazzan ^{1
,
,},
Mohammed Alsharafi ^2,3

1.
Department of Mathematics, Science Faculty, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Sana'a University, Sana'a, Yemen
3.
Department of Mathematics, Faculty of Arts and Science, Yildiz Technical University, Esenler 34220, Istanbul, Turkey

Received: 26 April 2026 Revised: 29 July 2026 Accepted: 06 August 2026 Published: 13 August 2026
MSC : 05C09, 05C50, 05C92, 92E10

Degree-based topological indices play an important role in chemical graph theory and network analysis, where structural heterogeneity is encoded through vertex degrees and aggregated over edges. In this paper, we introduce the elliptic Dharwad index (ED) of a simple graph $\mathcal{I}$,
$ ED(\mathcal{I}) = \sum\limits_{uv\in E(\mathcal{I})}(d(u)+d(v))\sqrt{d(u)^3+d(v)^3}, $
and investigate its behavior under several fundamental graph operations and extremal constraints. We establish sharp lower and upper bounds for $ED(\mathcal{I}+\mathcal{J})$ and $ED(\mathcal{I}\odot\mathcal{J})$ in terms of the orders, sizes, and extremal degrees of the factors, together with complete equality characterizations. We also derive exact edge-decomposition formulas and degree-based bounds for the Cartesian product $\mathcal{I}\square\mathcal{J}$ and the tensor product $\mathcal{I}\otimes\mathcal{J}$, including an extension to $k$-fold Cartesian products. As further applications, we obtain closed forms for several standard graph families arising from joins and corona products, including paths, cycles, and complete graphs. On the extremal side, we prove that $ED$ is strictly increasing under edge addition, determine sharp girth-based lower bounds for unicyclic graphs and connected graphs with a prescribed girth, and establish the best possible upper bound for connected graphs of order $\mathfrak{s}$ with exactly $\mathfrak{p}$ pendant vertices, showing that the unique extremal graph is $K_{\mathfrak{s}, \mathfrak{p}}$. To complement the theoretical results, we examine $ED$ in an exploratory univariate Quantitative Structure Property Relationship (QSPR) study on a dataset of 89 aromatic carboxylic acid, ester, and related benzenoid compounds, where the descriptor shows its clearest standalone signal for volumetric and size-related physicochemical properties. Overall, the results show that $ED$ admits exact evaluation on important graph classes, satisfies sharp extremal principles, and has meaningful descriptive utility in chemical graph applications.
- elliptic Dharwad index,
- degree-based topological indices,
- graph operations,
- join,
- corona product,
- Cartesian product,
- tensor product,
- extremal graphs,
- girth,
- pendant vertices,
- QSPR
Citation: Suha Wazzan, Mohammed Alsharafi. The elliptic Dharwad index for structural characterizations of extremal graphs and QSPR applications[J]. AIMS Mathematics, 2026, 11(8): 24839-24876. doi: 10.3934/math.2026999

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Abstract

Degree-based topological indices play an important role in chemical graph theory and network analysis, where structural heterogeneity is encoded through vertex degrees and aggregated over edges. In this paper, we introduce the elliptic Dharwad index (ED) of a simple graph $\mathcal{I}$,

$ ED(\mathcal{I}) = \sum\limits_{uv\in E(\mathcal{I})}(d(u)+d(v))\sqrt{d(u)^3+d(v)^3}, $

and investigate its behavior under several fundamental graph operations and extremal constraints. We establish sharp lower and upper bounds for $ED(\mathcal{I}+\mathcal{J})$ and $ED(\mathcal{I}\odot\mathcal{J})$ in terms of the orders, sizes, and extremal degrees of the factors, together with complete equality characterizations. We also derive exact edge-decomposition formulas and degree-based bounds for the Cartesian product $\mathcal{I}\square\mathcal{J}$ and the tensor product $\mathcal{I}\otimes\mathcal{J}$, including an extension to $k$-fold Cartesian products. As further applications, we obtain closed forms for several standard graph families arising from joins and corona products, including paths, cycles, and complete graphs. On the extremal side, we prove that $ED$ is strictly increasing under edge addition, determine sharp girth-based lower bounds for unicyclic graphs and connected graphs with a prescribed girth, and establish the best possible upper bound for connected graphs of order $\mathfrak{s}$ with exactly $\mathfrak{p}$ pendant vertices, showing that the unique extremal graph is $K_{\mathfrak{s}, \mathfrak{p}}$. To complement the theoretical results, we examine $ED$ in an exploratory univariate Quantitative Structure Property Relationship (QSPR) study on a dataset of 89 aromatic carboxylic acid, ester, and related benzenoid compounds, where the descriptor shows its clearest standalone signal for volumetric and size-related physicochemical properties. Overall, the results show that $ED$ admits exact evaluation on important graph classes, satisfies sharp extremal principles, and has meaningful descriptive utility in chemical graph applications.

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