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On Multiplicative Y -index of Product Graphs and Polycyclic Compounds

This paper introduces a novel multiplicative Y-index, derives its exact value for the tensor product of graphs and establishes upper bounds for various other graph products in terms of existing degree-based indices, and applies these results to compute the index for several polycyclic chemical compounds.

Original authors: Shivani Rai, Biswajit Deb, Sourav Mondal

Published 2026-10-27

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Shivani Rai, Biswajit Deb, Sourav Mondal

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Chemical compounds are often described by their formulas, which tell us how many atoms of each element are present. However, two substances can share the exact same formula while having completely different shapes and behaviors, a phenomenon known as structural isomerism. To distinguish between these look-alike molecules and predict how they will behave, scientists turn to a field called chemical graph theory. In this approach, a molecule is translated into a simple map: the atoms become dots, and the chemical bonds connecting them become lines. By studying the patterns of these connections, researchers can assign a single number to the entire structure. These numbers, known as topological indices, act as a fingerprint for the molecule's shape, often revealing hidden details about its boiling point, energy, or stability without needing to run a physical experiment in a lab.

For decades, scientists have developed various ways to calculate these fingerprints, focusing on how many connections each atom has. A newer approach involves looking at the fourth power of these connection counts, a method that has shown promise in predicting specific physical properties like the energy required to turn a liquid into a gas. Building on this, a team of researchers has introduced a new variation called the multiplicative Y-index. Instead of adding up these values, this method multiplies them together, creating a single, massive number that represents the entire molecular architecture. The researchers focused on two main tasks: first, they wanted to see how this new number behaves when two molecular maps are combined to form a larger, more complex structure; second, they applied their new method to calculate this index for several real-world chemical compounds that are important in industry and nanotechnology.

The study began by examining how molecular maps are constructed. Just as architects build large buildings by combining smaller, standard modules, chemists often create complex molecules by joining simpler ones together. The researchers looked at several specific rules for joining these molecular maps, such as the tensor product, where connections are made only when both original maps have a link at the same time, and the Cartesian product, which is more like laying one map on top of another and connecting matching points. The team derived precise mathematical rules to determine the new multiplicative Y-index for these combined structures. They found that for one specific type of combination, the index of the new, larger molecule could be calculated directly from the indices of the two original pieces, simply by raising the original numbers to certain powers and multiplying them. For other types of combinations, they established strict upper limits, showing the maximum possible value the index could reach based on the properties of the starting molecules. They also identified the specific conditions under which these maximum values are actually achieved, which happens only when the original molecular maps are perfectly uniform in their structure.

Having established these rules for combining shapes, the researchers turned their attention to actual chemical substances that appear in nature and industry. They applied their new calculation method to four distinct types of complex, ring-based structures. The first group was polycyclic aromatic hydrocarbons, a family of chemicals formed during the incomplete burning of fuels like coal and oil, which are known for their stability and presence in the environment. The second was a jagged rectangular system made of hexagonal cells, resembling a honeycomb with uneven edges. The third was a star-shaped molecule called a nanostar dendrimer, which is an artificial structure used in advanced technology for its ability to collect energy and interact with light. The final group was a metal-organic network, a sturdy framework used in high-temperature applications and catalysis. For each of these complex systems, the team calculated the exact value of the multiplicative Y-index, providing a precise numerical description of their structural complexity.

The work provides a new toolkit for understanding how the shapes of molecules relate to their physical properties. By defining this multiplicative index and showing how it behaves when molecular structures are combined, the researchers have offered a way to predict the characteristics of new, complex materials before they are even built. The study confirms that for certain regular, uniform structures, the relationship between the parts and the whole is exact and predictable. For more irregular combinations, the researchers have set clear boundaries on what is possible. This mathematical clarity helps bridge the gap between the abstract geometry of a molecule and its real-world behavior, offering a clearer path for designing new materials in fields ranging from energy storage to medical sensing.

Technical Summary: On Multiplicative Y-index of Product Graphs and Polycyclic Compounds

Problem Statement
Topological indices are numerical parameters used to analyze the chemical significance of molecular structures, often serving as descriptors for physico-chemical properties. While various degree-based indices (such as the Zagreb indices, Forgotten index, and the recently introduced Y-index) have been extensively studied for their additive forms, there is a need to extend these concepts to multiplicative variants. Specifically, this work addresses the lack of explicit expressions and bounds for the multiplicative Y-index ( $\Pi_Y$ ) in the context of product graphs and complex polycyclic chemical compounds. The paper seeks to define this novel index, derive its exact values for tensor products, establish upper bounds for other standard graph products, and compute its values for specific polycyclic aromatic and metal-organic networks.

Methodology
The study employs graph theory and algebraic manipulation to derive mathematical expressions for the multiplicative Y-index, defined as:
$\Pi_Y(\Gamma) = \prod_{u \in V(\Gamma)} d(u)^4$
where $d(u)$ is the degree of vertex $u$ .

The methodology proceeds in three stages:

Theoretical Derivation: The authors utilize the Arithmetic Mean-Geometric Mean (AM-GM) inequality (Lemma 1) to establish upper bounds for the multiplicative Y-index of various graph products. They analyze the degree functions of vertices in product graphs (Tensor, Cartesian, Lexicographic, Strong, Disjunction, and Symmetric Difference) as defined in Table 1.
Exact Computation for Tensor Products: For the tensor product ( $\Gamma_1 \otimes \Gamma_2$ ), the degree of a vertex $(u_1, u_2)$ is the product of the degrees of the factor vertices ( $d(u_1)d(u_2)$ ). The authors derive an exact formula relating $\Pi_Y(\Gamma_1 \otimes \Gamma_2)$ to the multiplicative Y-indices of the factor graphs.
Application to Chemical Structures: The derived formulas and bounds are applied to specific molecular graphs representing polycyclic compounds. The authors partition the vertex sets of these graphs based on vertex degrees (e.g., $V_1, V_2, V_3$ $V_{1}, V_{2}, V_{3}$ ) to compute the exact $\Pi_Y$ $Π_{Y}$ values for:
- Polycyclic Aromatic Hydrocarbons ( $PAH_n$ )
- Jagged Rectangular Benzenoid Systems ( $B_{m,n}$ )
- Polycyclic Aromatic Nanostar Dendrimers ( $D_1[n]$ )
- Polycyclic Metal Organic Networks ( $TM\text{-}TCNB$ )

Key Contributions and Results

Definition: The paper formally defines the multiplicative Y-index ( $\Pi_Y$ ) as the product of the fourth powers of the vertex degrees.
Tensor Product Formula: An exact expression is provided for the tensor product of two graphs $\Gamma_1$ and $\Gamma_2$ with orders $n_1$ and $n_2$ :
$\Pi_Y(\Gamma_1 \otimes \Gamma_2) = [\Pi_Y(\Gamma_1)]^{n_2} [\Pi_Y(\Gamma_2)]^{n_1}$
Upper Bounds for Other Products: The authors derive upper bounds for the multiplicative Y-index of the Cartesian, Lexicographic, Strong, Disjunction, and Symmetric Difference products. These bounds are expressed in terms of:
- The orders ( $n_1, n_2$ ) and sizes ( $m_1, m_2$ ) of the factor graphs.
- The First Zagreb Index ( $M_1$ ), Forgotten Index ( $F$ ), and Y-index ( $Y$ ) of the factor graphs.
- The equality conditions for these bounds are characterized: equality holds if and only if both factor graphs are regular.
Chemical Applications: Explicit formulas for $\Pi_Y$ $Π_{Y}$ are calculated for four classes of polycyclic compounds:
- $PAH_n$ : $81^{6n^2}$
- $B_{m,n}$ : $16^{2m+4n+2} \times 81^{4mn+2m-2n-4}$
- $D_1[n]$ : $16^{12 \times 2n - 7} \times 81^{4 \times 2n - 3}$
- $TM\text{-}TCNB$ : $16^{24n+8} \times 81^{76n-20} \times 256^{4n}$

Significance and Claims
The paper claims that the derived expressions allow for a better understanding of how topological indices of product graphs relate to the indices of their component building blocks. The authors note that while the additive Y-index has shown strong correlation with physico-chemical properties (such as acentric factor, entropy, and enthalpy of vaporization) in previous literature, this work extends that investigation to the multiplicative domain.

The study positions itself as a foundational mathematical exploration. It asserts that the computation of these new variants for different product graphs and chemical structures provides a reference for future mathematical research. The authors modestly frame the work as an extension of existing index theory, suggesting future directions include determining lower bounds and characterizing minimal graphs, as well as applying these results to other important classes of chemical compounds and nanostructures. The paper does not claim to have experimentally validated these indices against new chemical data, but rather provides the theoretical framework and exact values for their calculation.

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