Computational insights into zinc silicate MOF structures: topological modeling, structural characterization and chemical predictions

Abstract

Metal-organic frameworks (MOFs) play a pivotal role in modern material science, offering unique properties such as flexibility, substantial pore space, distinctive structure, and large surface area. Recently, zinc-based MOFs have attracted significant attention, particularly in the biomedical arena, owing to their versatile applications in drug delivery, biosensing, and cancer imaging. However, there remains a crucial need to explore and understand the structural properties of zinc silicate-based MOFs to fully exploit their potential in various applications. The objective of this study is to address this need by employing topological modeling techniques to characterize zinc silicate networks. Utilizing connection number concept of chemical graph theory and novel AL molecular descriptors, we aim to investigate the structural intricacies of these MOFs. More precisely, zinc silicate-based MOF networks are topologically modeled via novel AL topological indices, and derived mathematical closed form formulae for them. By comparing experimental and calculated values and constructing linear regression models, the predictive capabilities of the proposed descriptors are evaluated. Specifically, the performance of derived topological indices against the physico-chemical properties of octane isomers is assessed, which provide valuable insights into their predictive potential. The findings of this study demonstrated the potential of novel AL indices in predicting a wide range of important physico-chemical properties, further enhancing their practicality in materials science and beyond.

Introduction

Metal organic frameworks (MOFs) are unique classes of porous chemical materials characterized by a high pore volume, a unique structure, a huge surface area, and exceptional chemical stability. The broad range of applications of MOFs, such as environmental hazards, biocompatibility, toxicity, and biomedical applications, as well as heterogeneous catalysis, gas purification, and division, make them distinct¹. Their vast surface area, elevated porosity, adaptable structures, and simple chemical functionalization have made them attractive choices for pharmaceutical delivery, heterogeneous catalysts, nitric oxide storage, imaging and sensing, and illness diagnosis. It is interesting to see how the research and application of versatile MOFs have recently found a desirable and probable location in the field of health care². In biological applications, harmless MOFs are believed to work better than hazardous MOFs, especially in systems for delivering medicines. Among themost crucial elements of this industry's success is drug delivery^3,4. As shown in Fig. 1, the MOF is composed of organic molecules and ionized metal linkers⁵, a substance that is extremely permeable and has a large surface area.

Fig. 1

Structure of MOFs.

Graph theory offers a plenty of valuable tools for chemists and materials scientists, notably the topological indices. Using graph theory principles, molecular compounds are commonly represented through molecular graphs, which depict an element's structure as defined in chemistry. Chemical bonds are represented by edges in these graphs, while atoms are represented by vertices. The interdisciplinary field of cheminformatics integrates details, chemistry, and mathematics technology to explore the physical properties of chemical compounds, marking a significant advancement in scientific research. Cheminformatics provides a framework for mathematically modeling molecules based on their physical properties, such as boiling point etc. Milan Randic and NenadTrinajstic were among the first to explore chemical graph theory⁶. The process of transforming chemical information into meaningful numerical data is referred to as molecular descriptors or topological indices. These indices, which quantify various characteristics like connectivity, boiling point, stability, and melting point, are numeric parameters associated with molecular graphs representing chemical compounds. Topological indices seize the topology of the structures of molecules and often exhibit mathematical properties as graph invariants. They play a key role in quantitative structure–activity relationship (QSAR) studies, offering insights into the biological properties based on chemical structures.

While researching the boiling point of paraffin in 1947, Wiener presented the idea of the topological index, which he named as the Wiener index⁷. M. Darafsheh⁸ devised various methods for computing the Wiener index and Szeged index across multiple graphs. A. Ayache and A. Alameri⁹ computed topological indices for mk-graphs. Wei Gao et al.¹⁰ explored specific eccentricity-version topological indices within the cycloalkane family. Ullah and Zaman conducted extensive investigations into degree-based topological descriptors across diverse molecular structures^{11,12,13,14,15,16,17,18,19,20,21,22,23}. For further inquiries into the topological characterization of molecular structures and microstructures, references^{24,25,26,27,28,29,30,31,32} can be consulted.

The original inventor of the degree-based topological index was Milan Randic⁶.The degree sum of the end vertices of the edges in a graph G is the definition of the first Zagreb index, i.e.${\text{M}}_{1}\left(\text{G}\right)=\sum_{\text{uv}\in \left(\text{G}\right)}{(\text{d}}_{\text{u}}{+\text{ d}}_{\text{v}})$. Ivan Gutman introduced the second version of Zagreb index as the degree product of end vertices of all edges of the under consideration graph $\text{G}$³³, i.e. ${\text{M}}_{2}\left(\text{G}\right)=\sum_{\text{uv}\in \left(\text{G}\right)}{(\text{d}}_{\text{u }}\times {\text{d}}_{\text{v}})$. In 1998, Estrade et al.³⁴ Proposed $\text{ABC}$ index and defined it as,$\text{ABC}\left(\text{G}\right)={\sum }_{uv\in (G)}\sqrt{\frac{{d}_{u}+{d}_{v}-2}{{d}_{u}\times {d}_{v}}}$. In 2010, Gorbani et al.³⁵ proposed ${\text{ABC}}_{4}$ index and defined it as,${\text{ABC}}_{4}=\sqrt{\frac{{s}_{u}+{s}_{v }-2}{{s}_{u}\times {s}_{v}}}$.Furtula et al. Introduced another Zagreb type index in 2010,defined as,$\text{AZI}\left(\text{G}\right)=\sum_{uv\in E(G)}{(\frac{{d}_{u}{d}_{v}}{{d}_{u}+{d}_{v}-2})}^{3}$. The prediction power is this index is better than $\text{ABC}$ index. Vukievi³⁶introduced the first $\text{GA}$ index as,$\text{GA}\left(\text{G}\right)=\sum_{\text{uv}\in \text{E}(\text{G})}\frac{2\sqrt{{d}_{u}\times {d}_{v}}}{{d}_{u}+{d}_{v}}$. The lower and upper limits on $\text{GA}$ index were explained by Das et al.³⁷. Recently, Graovac et al.³⁸ introduced ${\text{GA}}_{5}$ index which is expressed as, ${\text{GA}}_{5}=\sum_{\text{uv}\in \text{E}(\text{G})}\frac{2\sqrt{{S}_{u}{\times S}_{v}}}{{S}_{u}+{S}_{v}}$.

In mathematical chemistry, topological indices come in several forms, such as distance-based indices^{39,40,41,42,43,44}, vertex-degree-based indices^{45,46,47,48,49,50,51,52,53}, spectrum-based indices^{54,55,56,57,58}, and connection-based indices^{59,60,61,62,63,64}. Recently, new AL indices' possible utilisation as molecular descriptors was investigated in⁶⁵. A more thorough comprehension of the molecular network structures is offered by these descriptors.The Connection-based Multiplicative Zagreb Indices of dendrimer nanostars were proposed by Sattar et al. in 2021¹⁷. For further details on the topological characterization of zinc-based MOFs, readers are referred to^66,67,68.

Motivated by the importance and wide applications of zinc silicate-based MOFs, this study employs topological modeling techniques to characterize zinc silicate networks in order to explore and understand the structural properties of zinc silicate-based MOFs to fully exploit their potential in various applications. Utilizing connection number concept of chemical graph theory and novel AL molecular descriptors, we aim to investigate the structural intricacies of these MOFs.

To be more specific, zinc silicate-based MOF networks are modelled topologically using novel AL topological indices, and formulas in closed form are derived mathematically. The predictive potential of the suggested descriptors is assessed by building linear regression models and comparing computed and experimental results. In particular, the efficacy of generated topological indices in relation to the physico-chemical characteristics of octane isomers is evaluated, offering significant perspectives on their predictive capacity. The results of this work can greatly enhanced the usefulness of AL indices in materials science and other fields by demonstrating their ability to predict a broad range of significant physico-chemical properties.

As, the structural flexibility and substantial pore space of zinc silicate-based MOFs make them ideal candidates for encapsulating and releasing pharmaceutical compounds in a controlled manner. Our topological indices can predict the stability and efficiency of these MOFs in drug delivery applications, ensuring optimal design and performance. Also, by employing the novel AL indices, researchers can predict and enhance the interaction of MOFs with target molecules, improving the efficacy of biosensing devices in medical diagnostics and environmental monitoring. Furthermore, by applying our topological modeling techniques, we can predict the gas adsorption capacity and selectivity of zinc silicate-based MOFs, guiding the design of materials for energy and environmental applications.

Preliminaries

Given a graph G = (T (G), W (G)) with T (G) as the vertex set and W (G) as the edge set, the number of vertices that are two distances away from a given vertex is its connection number. If we assume that ε = jk, where j and k are edges within T (G), then the innovative AL indices have the following definition:

Definition 2.1

${AL}_{1}$ is expressed as follows if G = (T(G),W(G));

$${AL}_{1}\left(G\right)=\sum_{j\in T\left(G\right)}{Y}_{2}\left(j\right)\dots \dots \dots \dots \dots \dots \dots \dots$$

(1)

The CNs associated with vertex j are represented by ${Y}_{2}\left(j\right)$.

Definition 2.2

For graph G, the index ${AL}_{2}\left(G\right)$ is as follows:

$${AL}_{2}\left(G\right)= \sum_{\left(j,k\right)\in W\left(G\right)}\{{Y}_{2}(j)+{Y}_{2}\left(k\right)\}\dots \dots \dots \dots \dots \dots \dots \dots$$

(2)

where ${Y}_{2}\left(j\right)$ and ${Y}_{2}\left(k\right)$ are respectively the CNs of vertices ‘j’ and ‘k’.

Definition 2.3

For a graph G, then ${AL}_{3}\left(G\right)$ is formulated as;

$${AL}_{3}\left(G\right)= \sum_{\left(j,k\right)\in W\left(G\right)}{Y}_{2}\left(j\right){Y}_{2}\left(k\right)\dots \dots \dots \dots \dots \dots \dots \dots$$

(3)

where ${Y}_{2}\left(j\right)$ and ${Y}_{2}\left(k\right)$ are respectively the CNs of vertices ‘j’ and ‘k’.

Definition 2.4

For a graph G, then ${AL}_{4}\left(G\right)$ is formulated as;

$${AL}_{4}\left(G\right)= \sum_{\left(j,k\right)\in W\left(G\right)}\sqrt{{Y}_{2}\left(j\right){Y}_{2}\left(k\right)}\dots \dots \dots \dots \dots \dots \dots \dots$$

(4)

where ${Y}_{2}\left(j\right)$ and ${Y}_{2}\left(k\right)$ are respectively the CNs of vertices ‘j’ and ‘k’..

Definition 2.5

For a graph G, then ${AL}_{5}\left(G\right)$ is formulated as;

$${AL}_{5}\left(G\right)= \sum_{\left(a,b\right)\in W\left(G\right)}\frac{1}{\sqrt{{Y}_{2}\left(j\right){Y}_{2}\left(k\right)}}\dots \dots \dots \dots \dots \dots \dots \dots$$

(5)

where ${Y}_{2}\left(j\right)$ and ${Y}_{2}\left(k\right)$ are respectively the CNs of vertices ‘j’ and ‘k’..

Definition 2.6

For a graph G, then ${AL}_{6}\left(G\right)$ is formulated as;

$${AL}_{6}\left(G\right)= \sum_{\left(j,k\right)\in W\left(G\right)}\{\frac{{Y}_{2}\left(j\right)}{{Y}_{2}\left(k\right)}+\frac{{Y}_{2}\left(k\right)}{{Y}_{2}\left(j\right)}\}\dots \dots \dots \dots \dots \dots \dots \dots$$

(6)

where ${Y}_{2}\left(j\right)$ and ${Y}_{2}\left(k\right)$ are respectively the CNs of vertices ‘j’ and ‘k’..

Definition 2.7

For a graph G, then ${AL}_{7}\left(G\right)$ is formulated as;

$${AL}_{7}\left(G\right)= \sum_{\left(j,k\right)\in W\left(G\right)}\{\frac{{Y}_{2}\left(j\right){Y}_{2}\left(k\right)}{{Y}_{2}\left(j\right)+{Y}_{2}(k)}\}\dots \dots \dots \dots \dots \dots \dots \dots$$

(7)

where ${Y}_{2}\left(j\right)$ and ${Y}_{2}\left(k\right)$ are respectively the CNs of vertices ‘j’ and ‘k’..

Definition 2.8

For a graph G, then ${AL}_{8}\left(G\right)$ is formulated as;

$${AL}_{8}\left(G\right)= \sum_{\left(j,k\right)\in W\left(G\right)}\frac{2}{{Y}_{2}\left(j\right)+{Y}_{2}(k)}\dots \dots \dots \dots \dots \dots \dots \dots$$

(8)

where ${Y}_{2}\left(j\right)$ and ${Y}_{2}\left(k\right)$ are respectively the CNs of vertices ‘j’ and ‘k’..

Results and discussion

Mathematical formulation of novel AL indices for zinc silicate MOF structures

This section presents mathematical modeling and derivations of ${AL}_{1}$, ${AL}_{2}$, ${AL}_{3}$, ${AL}_{4}$, ${AL}_{5}$, ${AL}_{6}$,${AL}_{7} and{ AL}_{8}$ for the zinc silicate MOF structures of growth h. Given a molecular network of zinc silicate with growth h, denoted by τ = ZnSi(h), the cardinality of its edges and vertices is 85h + 55 and 70h + 46, respectively. Figures 2, 3 and 4 display the ZnSi connection-based structure for h = 1, 2 and 3. To keep things simple, ZnSi's structure is broken down into three parts: The Root, Stem, and Leaf structures.

Fig. 2

Structure of the ZnSi graph at h = 1.

Fig. 3

Structure of the ZnSi graph at h = 2.

Fig. 4

Structure of the ZnSi graph at h = 3.

• 1. Root structure: four hexagons joined together so that their common vertex has Connection Number (CN) 8 and their two outer vertices have CN 2 for h $\ge 1$ constitute a construction known as a Root structure.

• 2. Stem structure: a four-hexagon structure called a "Stem structure" is one that is connected such that the number CN 8 appears on its common vertex and the number CN 2 appears for h $\ge 2$ on its outer vertex.

• 3. Leaf structure: referred to as the Leaf structure, a hexagon is one that is not a part of the Stem and Root structures of hexagons for h $\ge 1$.

Theorem 3.1

Let the graph of growth for zinc silicate, $\tau$= ZnSi, have a growth rate of $h\ge 1$. The ${AL}_{1}$ index is then provided by

$${AL}_{1}\left(\tau \right)=82h+50$$

Proof

Here ${Y}_{2}\left(a\right)$, indicates CNs. Equation (1) and Table 1 are used to obtain

Table 1 ${Y}_{2}\left(a\right)$ denotes the CNs.

$${AL}_{1}\left(\tau \right)=\sum_{j\in T\left(\tau \right)}{Y}_{2}\left(a\right)$$

$${AL}_{1}\left(\tau \right)={Y}_{2}\left(2\right)+{Y}_{2}\left(3\right)+{Y}_{2}\left(4\right)+{Y}_{2}\left(5\right)+{Y}_{2}\left(6\right)+{Y}_{2}\left(8\right)$$

$$=\left(2h+6\right)+\left(16h+16\right)+\left(48h+16\right)+(8h+8)+(6h+2)+(2h+2)$$

$$=82h+50$$

Theorem 3.2

Let the growth rate on the graph of zinc silicate, $\tau =$ ZnSi, be $h\ge 1$. The ${AL}_{2}$ index is thus given by

$${AL}_{2}\left(\tau \right)=876h+500$$

Proof

Here ${Y}_{2}\left(j\right)$ and ${Y}_{2}\left(k\right)$, represent the CNs of vertices 'j' and 'k,' respectively. Equation (2) and Table 2 are used to obtain

Table 2 Partition edges and their cardinality $\left|{D}_{\left(a,b\right)}(\tau )\right|$.

$${AL}_{2}\left(\tau \right)= \sum_{\left(j,k\right)\in W\left(\tau \right)}\{{Y}_{2}(j)+{Y}_{2}\left(k\right)\}$$

$$\begin{aligned}{AL}_{2}\left(\tau \right)& = \left|{D}_{\left(\text{2,3}\right)}(\tau )\right|\left\{{Y}_{2}\left(2\right)+{Y}_{2}\left(3\right)\right\}+\left|{D}_{\left(\text{3,3}\right)}(\tau )\right|\left\{{Y}_{2}\left(3\right)+{Y}_{2}\left(3\right)\right\}+\left|{D}_{\left(\text{3,4}\right)}(\tau )\right|\left\{{Y}_{2}\left(3\right)+{Y}_{2}\left(4\right)\right\}\\ & \quad +\left|{D}_{\left(\text{3,5}\right)}(\tau )\right|\left\{{Y}_{2}\left(3\right)+{Y}_{2}\left(5\right)\right\}+\left|{D}_{\left(\text{4,4}\right)}(\tau )\right|\left\{{Y}_{2}\left(4\right)+{Y}_{2}\left(4\right)\right\}+\left|{D}_{\left(\text{4,5}\right)}(\tau )\right|\left\{{Y}_{2}\left(4\right)+{Y}_{2}\left(5\right)\right\}\\ & \quad +\left|{D}_{\left(\text{4,6}\right)}(\tau )\right|\{{Y}_{2}(4)+{Y}_{2}\left(6\right)\}+\left|{D}_{\left(\text{5,8}\right)}(\tau )\right|\left\{{Y}_{2}\left(5\right)+{Y}_{2}\left(8\right)\right\}+\left|{D}_{\left(\text{6,6}\right)}(\tau )\right|\{{Y}_{2}(6)+{Y}_{2}\left(6\right)\}1\end{aligned}$$

$$\begin{aligned}&=\left(4h+12\right)\left\{2+3\right\}+\left(6h+2\right)\left\{3+3\right\}+\left(12h+4\right)\left\{3+4\right\}+\left(4h+12\right)\left\{3+5\right\}+\left(42h+14\right)\left\{4+4\right\}\\ & \quad+\left(12h+4\right)\left\{4+5\right\}+(12h+4)\{4+6\}+(8h+8)\{5+8\}+(3h+1)\{6+6\}\end{aligned}$$

$$\begin{aligned}& =\left(4h+12\right)\left\{5\right\}+\left(6h+2\right)\left\{6\right\}+\left(12h+4\right)\left\{7\right\}+\left(4h+12\right)\left\{8\right\}+\left(42h+14\right)\left\{8\right\}\\ &\quad+\left(12h+4\right)\left\{9\right\}+(12h+4)\{10\}+(8h+8)\{13\}+(3h+1)12\end{aligned}$$

$$=20h+60+36h+12+84h+28+32h+96+336h+112+108h+36+120h+40+104h+104+36h+12$$

$$=876h+500$$

Theorem 3.3

On the zinc silicate graph, $\tau$= ZnSi, and let $h\ge 1$ represent the growth rate. Thus, the ${AL}_{3}$ index is represented by

$${AL}_{3}\left(\tau \right)=1910h+1074$$

Proof

${Y}_{2}\left(j\right)$ and ${Y}_{2}\left(k\right)$, respectively, stand for the CNs of vertices ‘j’ and ‘k’ in this case. By using Table 2 and Eq. (3), we get

$${AL}_{3}\left(\tau \right)= \sum_{\left(j,k\right)\in W\left(\tau \right)}\{{Y}_{2}(j){Y}_{2}\left(k\right)\}$$

$$\begin{aligned}{AL}_{3}\left(\tau \right) & = \left|{D}_{\left(\text{2,3}\right)}(\tau )\right|\left\{{Y}_{2}\left(2\right){Y}_{2}\left(3\right)\right\}+\left|{D}_{\left(\text{3,3}\right)}(\tau )\right|\left\{{Y}_{2}\left(3\right){Y}_{2}\left(3\right)\right\}+\left|{D}_{\left(\text{3,4}\right)}(\tau )\right|\left\{{Y}_{2}\left(3\right){Y}_{2}\left(4\right)\right\} \\ & \quad +\left|{D}_{\left(\text{3,5}\right)}(\tau )\right|\left\{{Y}_{2}\left(3\right){Y}_{2}\left(5\right)\right\} +\left|{D}_{\left(\text{4,4}\right)}(\tau )\right|\left\{{Y}_{2}\left(4\right){Y}_{2}\left(4\right)\right\}+\left|{D}_{\left(\text{4,5}\right)}(\tau )\right|\left\{{Y}_{2}\left(4\right){Y}_{2}\left(5\right)\right\}\\ & \quad +\left|{D}_{\left(\text{4,6}\right)}(\tau )\right|\{{Y}_{2}(4){Y}_{2}\left(6\right)\}+\left|{D}_{\left(\text{5,8}\right)}(\tau )\right|\left\{{Y}_{2}\left(5\right){Y}_{2}\left(8\right)\right\} +\left|{D}_{\left(\text{6,6}\right)}(\tau )\right|\{{Y}_{2}(6){Y}_{2}\left(6\right)\}\end{aligned}$$

$$\begin{aligned}& =\left(4h+12\right)\left\{2\times 3\right\}+\left(6h+2\right)\left\{3\times 3\right\}+\left(12h+4\right)\left\{3\times 4\right\}+\left(4h+12\right)\left\{3\times 5\right\} +\left(42h+14\right)\left\{4\times 4\right\}\\ & \quad+\left(12h+4\right)\left\{4\times 5\right\}+(12h+4)\{4\times 6\}+(8h+8)\{5\times 8\}+(3h+1)\{6\times 6\}\end{aligned}$$

$$\begin{aligned}& =\left(4h+12\right)\left\{6\right\}+\left(6h+2\right)\left\{9\right\}+\left(12h+4\right)\left\{12\right\}+\left(4h+12\right)\left\{15\right\} +\left(42h+14\right)\left\{16\right\}\\ & \quad+\left(12h+4\right)\left\{20\right\}+(12h+4)\{24\}+(8h+8)\{40\}+(3h+1)\{36\}\end{aligned}$$

$$=24h+72+54h+18+144h+48+60h+180+672h+224+240h+80+288h+96+320h+320+108h+36$$

$$=1910h+1074$$

Theorem 3.4

Let $h\ge 1$ represent the growth rate on the zinc silicate graph with $\tau$ = ZnSi. Accordingly, the ${AL}_{4}$ index is represented by

$${AL}_{4}\left(\tau \right)=433.9092h+245.8076$$

Proof

The CNs of vertices ‘j’ and ‘k’ in this instance are represented, respectively, by ${Y}_{2}\left(j\right)$ and ${Y}_{2}\left(k\right)$. By using Table 2 and Eq. (4), we get

$${AL}_{4}\left(\tau \right)= \sum_{\left(j,k\right)\in W\left(\tau \right)}\sqrt{{Y}_{2}\left(j\right){Y}_{2}\left(k\right)}$$

$$\begin{aligned}{AL}_{4}\left(\tau \right)& =\left|{D}_{\left(\text{2,3}\right)}(\tau )\right|\sqrt{\left\{{Y}_{2}\left(2\right){Y}_{2}\left(3\right)\right\}}+\left|{D}_{\left(\text{3,3}\right)}(\tau )\right|\sqrt{\left\{{Y}_{2}\left(3\right){Y}_{2}\left(3\right)\right\}}+\left|{D}_{\left(\text{3,4}\right)}(\tau )\right|\sqrt{\left\{{Y}_{2}\left(3\right){Y}_{2}\left(4\right)\right\}}\\ & \quad +\left|{D}_{\left(\text{3,5}\right)}(\tau )\right|\sqrt{\left\{{Y}_{2}\left(3\right){Y}_{2}\left(5\right)\right\}}+\left|{D}_{\left(\text{4,4}\right)}(\tau )\right|\sqrt{\left\{{Y}_{2}\left(4\right){Y}_{2}\left(4\right)\right\}}+\left|{D}_{\left(\text{4,5}\right)}(\tau )\right|\sqrt{\left\{{Y}_{2}\left(4\right){Y}_{2}\left(5\right)\right\}}\\ & \quad+\left|{D}_{\left(\text{4,6}\right)}(\tau )\right|\sqrt{\{{Y}_{2}(4){Y}_{2}\left(6\right)\}}+\left|{D}_{\left(\text{5,8}\right)}(\tau )\right|\sqrt{\left\{{Y}_{2}\left(5\right){Y}_{2}\left(8\right)\right\}}+\left|{D}_{\left(\text{6,6}\right)}(\tau )\right|\sqrt{\{{Y}_{2}(6){Y}_{2}\left(6\right)\}}\end{aligned}$$

$$\begin{aligned} & =\left(4h+12\right)\sqrt{\left\{2\times 3\right\}}+\left(6h+2\right)\sqrt{\left\{3\times 3\right\}}+\left(12h+4\right)\sqrt{\left\{3\times 4\right\}}+\left(4h+12\right)\sqrt{\left\{3\times 5\right\}} +\left(42h+14\right)\sqrt{\left\{4\times 4\right\}}\\ & \quad+\left(12h+4\right)\sqrt{\left\{4\times 5\right\}}+(12h+4)\sqrt{\{4\times 6\}}+(8h+8)\sqrt{\{5\times 8\}}+(3h+1)\sqrt{\{6\times 6\}}\end{aligned}$$

$$\begin{aligned} & =\left(4h+12\right)\sqrt{6}+\left(6h+2\right)\sqrt{9}+\left(12h+4\right)\sqrt{12}+\left(4h+12\right)\sqrt{15}+\left(42h+14\right)\sqrt{16}\\ & \quad +\left(12h+4\right)\sqrt{20}+(12h+4)\sqrt{24}+(8h+8)\sqrt{40}+(3h+1)\sqrt{36}\end{aligned}$$

$$\begin{aligned} & =\left(4h+12\right)2.4495+\left(6h+2\right)3+\left(12h+4\right)3.4641+\left(4h+12\right)3.8730+\left(42h+14\right)4+\left(12h+4\right)4.4721 \\ & \quad +\left(12h+4\right)4.8990+\left(8h+8\right)6.3246+(3h+1)6\end{aligned}$$

$$\begin{aligned}& = 9.7980h+29.3940+18h+6+41.5692h+13.8564+15.4920h+46.4760+168h+56\\ & \quad+53.6652h+17.8884 +58.7880h+19.5960+50.5968h+50.5968++18h+6\end{aligned}$$

$$=433.9092h+245.8076$$

Similarly, all the other AL indices, namely, AL₄, AL₅, AL₆, AL₇ and AL₈ are derived mathematically. Table 3 below summarizes the obtained results for all the 8 AL indices.

Table 3 Mathematically derived closed form expressions for AL indices of zinc silicate (ZnSi).

Property prediction ability of the derived indices and discussion

Table 4 presents the numerically computed AL index values for Zinc Silicate MOFs, and Fig. 5 provides a graphical comparison of these indices. It is evident from both the table and the figure that AL₃ index attains the highest value for ZnSi. Furthermore, there is a clear trend of increasing index values with increasing values of h.

Table 4 Numerical values of the AL indices for zinc silicate (ZnSi) MOFs.

Fig. 5

Comparison of AL indices for ZnSi.

To estimate the predictive ability in order to assess the possible utility and chemical applicability of these AL indices, we employed Octane Isomers dataset together with the associated experimental properties. In what follows, a comprehensive investigation of the predictive abilities and chemical applicability of AL topological indices is performed.

To assess how well a topological index works in modeling physico-chemical properties, regression analysis serves as a powerful tool⁵³. Octane isomers, with their diverse structural characteristics including form, branching, and non-polar traits, offer an ideal test bed for such investigations. Their structural diversity ensures a wide range of experimental data availability, making octane isomers particularly suitable for statistical analysis. According to Randić and Trinajstić⁶⁹, correlating theoretical constants with the experimental physicochemical characteristics of octane isomers is crucial to evaluating an invariant's predictive power^70,71,72. In this work, we first calculated novel AL indices for 18 Octane Isomers and then correlated the calculated values with the experimental parameters to test the prediction power and practical usefulness of these indices. The experimental data were taken from the studies^53,73. Similar to the methods outlined in "Mathematical formulation of Novel AL indices for zinc silicate MOF structures", the calculations of AL indices for these isomers have already been accomplished in a recent work ⁷⁴.We have used the numerical values of AL indices for Octane Isomers obtained in⁷⁴.

Using regression analysis, we looked into seven physico-chemical characteristics for each of the eighteen octane isomers. Origin, MATLAB, and Excel softwares were used for analysis and visualization of the results.

The correlations between the physico-chemical properties of the Octane Isomers and the AL indices are shown in Figs. 6, 7, 8, 9, 10, 11 and 12 (presented in supplementary information), and the values of correlation co-efficient are shown in Table 5. Interestingly, our findings reveal intriguing correlations: AL₁ exhibits a robust connection (r = 0.86085) with the octane isomer entropy, while AL₁, AL₂, AL₃, AL₄, AL₅, and AL₇ show strong correlations (r = 0.83757, r = 0.86983, r = 0.79237, r = 0.8219, r = 0.79527 and r = 0.73139 respectively) with the Acentric Factor, with AL₂ having the strongest correlation among them. As can be seen, AL₂, AL₄, and AL₅ exhibit strong correlations (r = 0.72745, r = 0.71056, and r = 0.71138 respectively) with the HVAP, with AL₂ showing the strongest correlation among them. Moreover, AL₁, AL₂, AL₃, AL₄, and AL₅ show strong correlations (r = 0.76855, r = 0.791, r = 0.71274, r = 0.7683, and r = 0.74952 respectively) with the DHVAP, with AL₂ emerging as the most predictive index. However, the Boiling Point, Critical Temperature, and Critical Pressure of the octane isomers have weak associations with the eight unique AL indices.

Fig. 6

Correlation of AL₁ with physico-chemical properties.

Fig. 7

Correlation of AL₂ with physico-chemical properties.

Fig. 8

Correlation of AL₃ with physico-chemical properties.

Fig. 9

Correlation of AL₄ with physico-chemical properties.

Fig. 10

Correlation of AL₅ with physico-chemical properties.

Fig. 11

Correlation of AL₇ with physico-chemical properties.

Fig. 12

Correlation of AL₈ with physico-chemical properties.

Table 5 Values of the correlation coefficients.

The strong performance of certain AL indices in predicting the properties of octane isomers can be attributed to their inherent sensitivity to molecular structure variations. Specifically, AL indices are designed to capture the topological and connectivity features of molecules, which are directly related to their physical and chemical properties. For instance, AL₁'s robust connection with octane isomer entropy suggests that this index effectively captures the complexity and disorder within the molecular structure, which is a key component of entropy. The ability of AL indices to encapsulate such intricate structural details likely contributes to their strong predictive capabilities. The high correlations between AL indices and the Acentric Factor, particularly AL₂, indicate that these indices are sensitive to the asymmetry and deviations from ideal behavior in the molecules. The Acentric Factor is a measure of the non-sphericity of molecules, and the strong correlations suggest that AL indices can effectively represent such geometric and spatial characteristics. Similarly, the strong correlations with HVAP and DHVAP imply that AL indices can reflect intermolecular forces and the energy required for phase transitions. The HVAP and DHVAP are influenced by molecular interactions and bonding, which are well-represented by the topological features captured by AL indices.

However, the weaker associations with Boiling Point, Critical Temperature, and Critical Pressure might indicate that these properties are influenced by factors beyond the scope of topological indices alone, such as specific intermolecular interactions or external conditions that are not fully encapsulated by AL indices.

To sum up, the effectiveness of AL indices in predicting a range of physico-chemical properties underscores their utility in materials science, particularly for applications requiring detailed molecular insights. The development and refinement of such indices can lead to more accurate and comprehensive models for predicting molecular behavior, thus broadening their applicability across various scientific domains. While the properties of octane isomers are well-understood, our study leverages these principles to validate and showcase the effectiveness of novel AL indices. This validation is a crucial step in ensuring that these indices can be successfully applied to zinc silicate-based MOFs and other complex materials. By establishing a strong foundation with octane isomers, we enhance the robustness and applicability of AL indices to a broader range of materials in materials science and biomedicine.

Conclusion

In this study, the structural intricacies of the MOFs are investigated by utilizing connection number concept of chemical graph theory and novel AL molecular descriptors (AL₁, AL₂, AL₃, AL₄, AL₅, AL₆, AL₇ and AL₈). More precisely, zinc silicate-based MOF structures are topologically modeled via novel AL topological indices, and derived mathematical closed form formulae for them. By comparing experimental and calculated values and constructing linear regression models, we evaluated the predictive capabilities of the proposed descriptors. In particular, we evaluated the obtained topological indices' performance in relation to the physico-chemical characteristics of octane isomers, offering important information about their predictive capacity.

Our findings reveal significant correlations between the AL indices and the experimental properties. Notably, AL₁ exhibits a strong correlation (r = 0.86085) with entropy. Additionally, AL₁, AL₂, AL₃, AL₄, AL₅, and AL₇ show robust correlations with the Acentric Factor, with AL₂ having the highest correlation (r = 0.86983). AL₂ also stands out in predicting HVAP and DHVAP properties, showing strong correlations (r = 0.72745 and r = 0.791, respectively). These results underscore the predictive power of the AL indices, particularly AL₁ and AL₂, in capturing key structural features that influence the physico-chemical properties. The successful application of these novel AL indices to octane isomers provides a robust validation of their utility, paving the way for their application to more complex materials like zinc silicate-based MOFs. This study highlights the potential of AL indices to predict critical physico-chemical properties, enhancing our understanding of MOF structures and their suitability for diverse applications in materials science and biomedicine. However, the weaker associations with Boiling Point, Critical Temperature, and Critical Pressure might indicate that these properties are influenced by factors beyond the scope of topological indices alone, such as specific intermolecular interactions or external conditions that are not fully encapsulated by AL indices.

We recognize that MOFs have unique structural and functional characteristics that may require additional or different descriptors for comprehensive modeling. As part of our ongoing research, we plan to expand our set of indices and explore additional properties that are specifically relevant to MOFs. This will include properties related to pore size distribution, surface area, and framework flexibility, among others. Furthermore, future research could benefit from the integration of machine learning techniques to improve the predictive power of the AL indices. The accuracy and robustness of the predictions could be improved by training models on larger datasets that include a variety of MOF structures and physico-chemical properties.

Data availability

All data generated or analyzed during this study are included in this article.