1 Introduction

Since its discovery in 1984 in Kenya, AIDS has rapidly spread throughout numerous nations, resulting in a significant global impact. As of April 15, 2020, more than 1.97 million confirmed cases and over 125,000 fatalities have been reported worldwide, highlighting the urgent need for effective and secure medications and vaccines to treat and prevent this devastating disease [1]. One such medication used in the treatment of HIV is tenofovir, often administered alongside other antiretroviral drugs. Variants of tenofovir, including tenofovir dimer, tenofovir disoproxil, and tenofovir alafenamide, have shown promising results in managing the infection, particularly for high-risk individuals [2].

Medicinal science encompasses the study of pharmaceutical, chemical, medical, and biological aspects of medications, with a focus on understanding molecular structures and their associated topological indices [3,4,5,6]. Molecular structures can be represented as graphs, where atoms are represented as vertices, and bonds as edges. The field of Chemical Graph Theory plays a crucial role in unraveling the complexities of molecular structures [7,8,9,10].

Dual degree-based topological indices were first investigated in 1972 [11], marking a significant milestone in the study of chemical graphs [12,13,14]. Topological indices [15, 16], also known as molecular descriptors, are numerical values that provide insights into the chemical composition, reactivity, physical properties, or biological activity of a substance.

In this study, we delve into the examination of the chemical structures of HIV/AIDS medications using well-established degree-based topological indices. By representing the molecular structures as graphs, with elements as vertices and boundaries as edges, we aim to explore the relationship between these topological indices and the physical characteristics of HIV/AIDS drugs [17]. Furthermore, through rigorous quantitative structure–property relationship (QSPR) analysis [18,19,20], we demonstrate the strong correlation between these topological indices and the characteristics of the drugs [21].

The implications of our findings are significant for chemists and professionals in the pharmaceutical industry, as they offer a means to predict the characteristics of HIV/AIDS drugs without the need for extensive experimental testing. This study sheds light on the potential of topological indices as valuable tools in drug discovery and development, specifically in the context of HIV/AIDS medications. By leveraging these indices, researchers can expedite the identification and optimization of potential drug candidates, ultimately contributing to the ongoing efforts to combat this global health crisis. By delving into the intricate details of these chemical structures, namely Disoproxil, Alafenamide, and Dimer, and exploring the degrees of neighboring vertices, we can gain a more profound insight into their fundamental molecular attributes. These discoveries provide valuable insights for the field of chemical analysis and lay the foundation for additional research and exploration in drug discovery, molecular modeling, and other related scientific pursuits.

2 Basic notations and definitions about neighbourhood version topological descriptor

Let \(\chi (V, E)\) is a simple, connected graph where \(V(\chi )\) is its vertex set and \(E(\chi )\) is its edge set for any vertex \(\nu , \omega \in V(\chi )\) and for any edge \(\nu \omega \in E(\chi )\). The neighborhood degree sum of \(\nu \in V\), \(\omega \in V\) denotes as \(S_\nu\), \(S_\omega\) respectively. Where \(S_\nu =\sum _{\nu \in N_\chi (\nu )} d_\chi (\nu )\), \(N_\chi (\nu )\) be the set of vertices adjacent to \(\nu\). The neighbourhood version of the various indices were introduced [22, 23] and are defined as follows (Fig. 1).

Fig. 1
figure 1

Chemical Structure of Disoproxil

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2.1 Neighborhood first zagreb index

$$\begin{aligned} NM_{1}(\chi )=\sum _{vw\in E(\chi )}(S_{v}+S_{w}) \end{aligned}$$

2.2 Neighborhood second zagreb index

$$\begin{aligned} NM_{2}(\chi )=\sum _{vw\in E(\chi )}(S_{v} \times S_{w}) \end{aligned}$$

2.3 Neighborhood hyper zagreb index

$$\begin{aligned} HM_{N}(\chi )=\sum _{vw\in E(\chi )}(S_{v}+S_{w})^{2} \end{aligned}$$

2.4 Neighborhood redefined third zagreb index

$$\begin{aligned} ND_{6}(\chi )=\sum _{vw\in E(\chi )}(S_{v}S_{w})(S_{v}+S_{w}) \end{aligned}$$

2.5 Neighborhood harmonic index

$$\begin{aligned} {NH(\chi )= \sum _{vw\in E(\chi )}\frac{2}{(S_{v} + S_{w})}} \end{aligned}$$

2.6 Neighborhood forgotten index

$$\begin{aligned} {F_N(\chi )= \sum _{v\in V(\chi )}{S_{v}^3}} \end{aligned}$$

2.7 Neighborhood modified forgotten index

$$\begin{aligned} {F^*_N(\chi )= \sum _{v w\in E(\chi )}(S_{v}^2+S_{w}^2)} \end{aligned}$$
Fig. 2
figure 2

Chemical Structure of Alafenamide

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Table 1 The vertex partition of Disoproxil chemical structure
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Table 2 The edge partition of Disoproxil chemical structure
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3 Results and discussions

In this section, we have presented important results that relate to the chemical structures of three different compounds, namely Tenofovir alafenamide, disoproxil, and Tri-Poc Dimer. These compounds are widely used in the medical field, particularly in the treatment of HIV/AIDS (Fig. 2).

To investigate the physical properties of these compounds, we employed several useful degree-based topological indices. These indices were instrumental in the modeling of seven different physical properties, namely boiling point (BP), enthalpy of vaporization (E), molar volume (MV), flash point, polarizability, surface tension, and molar refractivity.

By analyzing the topological indices and physical properties, we were able to gain valuable insights into the molecular structures of these compounds. This information is crucial for understanding the behavior of these compounds in various chemical and biological systems, and can assist in the development of more effective drugs in the future.

In the pursuit of understanding the intricate chemical structures of Disoproxil, Alafenamide, and Dimer, a detailed analysis of the degrees of neighboring vertices was conducted. For Disoproxil, the vertex set cardinality was determined as \(\vert V(\chi _{n})\vert =34n+1\), resulting in the identification of six distinct vertex types with degrees spanning from 3 to 8, as presented in Table 1. Correspondingly, Table 2 offers a comprehensive overview of the diverse forms of edges encountered within the realm of neighborhood vertex degrees.

Similarly, for Alafenamide, the vertex set cardinality was determined as \(\vert V(\chi _{n})\vert =32n+1\), leading to the observation of six unique vertex types with degrees ranging from 3 to 8, as outlined in Table 3. In tandem, Table 4 provides an encompassing summary of the various manifestations of edges encountered within the purview of neighborhood vertex degrees.

Lastly, the investigation of Dimer involved calculations pertaining to the degrees of neighboring vertices in the graph representation. The cardinality of the vertex set for Dimer was determined as \(\vert V(\chi _{n})\vert =62n+1\), unveiling the presence of six distinct vertex types with degrees spanning from 3 to 8, as elucidated in Table 5. Furthermore, a comprehensive summary of the diverse forms of edges encountered within the context of neighborhood vertex degrees can be found in Table 6.

By delving into the intricate details of these chemical structures and exploring the degrees of neighboring vertices, a deeper understanding of their underlying molecular characteristics can be attained. These findings serve as valuable insights in the field of chemical analysis and pave the way for further research and exploration in drug discovery, molecular modeling, and related scientific endeavors.

Theorem 3.1

For the Disoproxil chemical structure,

  1. (1)

    The Neighborhood first zagreb index

    $$\begin{aligned} NM_{1}(\chi _{n})=390n-12, \end{aligned}$$
  2. (2)

    The Neighborhood second zagreb index

    $$\begin{aligned} NM_{2}(\chi _{n})=1080n-64, \end{aligned}$$
  3. (3)

    The Neighborhood Hyper-Zagreb index

    $$\begin{aligned} HM_{N}(\chi _{n})=4402n-248, \end{aligned}$$
  4. (4)

    The Neighborhood redefined third zagreb index

    $$\begin{aligned} ND_{6}(\chi _{n})=12690n-1014, \end{aligned}$$
  5. (5)

    The Neighborhood Harmonic index

    $$\begin{aligned} NH(\chi _{n})=\frac{499817}{72072}n-\frac{45197}{180180}, \end{aligned}$$
  6. (6)

    The Neighborhood Forgotten index

    $$\begin{aligned} F_N(\chi _{n})=5422n-350, \end{aligned}$$
  7. (7)

    The Neighborhood Modified Forgotten index

    $$\begin{aligned} F^*_N(\chi _{n})=2242n-233. \end{aligned}$$
Table 3 The vertex partition of Alafenamide chemical structure
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Table 4 The edge partition of Alafenamide chemical structure
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Theorem 3.2

For the Alafenamide chemical structure,

  1. (1)

    The Neighborhood first zagreb index

    $$\begin{aligned} NM_{1}(\chi _{n})=396n-12, \end{aligned}$$
  2. (2)

    The Neighborhood second zagreb index

    $$\begin{aligned} NM_{2}(\chi _{n})=1142n-64, \end{aligned}$$
  3. (3)

    The Neighborhood hyper zagreb index

    $$\begin{aligned} HM_{N}(\chi _{n})=4646n-248, \end{aligned}$$
  4. (4)

    The Neighborhood redefined third zagreb index

    $$\begin{aligned} ND_{6}(\chi _{n})=13878n-1007, \end{aligned}$$
  5. (5)

    The Neighborhood Harmonic index

    $$\begin{aligned} NH(\chi _{n})=\frac{154403}{24024}n-\frac{45197}{180180}, \end{aligned}$$
  6. (6)

    The Neighborhood Forgotten index

    $$\begin{aligned} F_N(\chi _{n})=5804n-350, \end{aligned}$$
  7. (7)

    The Neighborhood Modified Forgotten index

    $$\begin{aligned} F^*_N(\chi _{n})=2362n-120. \end{aligned}$$
Fig. 3
figure 3

Chemical Structure of Dimer

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Table 5 The vertex partition of Dimer chemical structure
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Table 6 The edge partition of Dimer chemical structure
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Theorem 3.3

For the Dimer chemical structure,

  1. (1)

    The Neighborhood First Zagreb index

    $$\begin{aligned} NM_{1}(\chi _{n})=718n-12, \end{aligned}$$
  2. (2)

    The Neighborhood Second Zagreb index

    $$\begin{aligned} NM_{2}(\chi _{n})=1990n-54, \end{aligned}$$
  3. (3)

    The Neighborhood hyper zagreb index

    $$\begin{aligned} HM_{N}(\chi _{n})=8112n-216, \end{aligned}$$
  4. (4)

    The Neighborhood redefined third zagreb index

    $$\begin{aligned} ND_{6}(\chi _{n})=23358n-744, \end{aligned}$$
  5. (5)

    The Neighborhood Harmonic index

    $$\begin{aligned} NH(\chi _{n})=\frac{189257}{15015}n-\frac{243}{770}, \end{aligned}$$
  6. (6)

    The Neighborhood Forgotten index

    $$\begin{aligned} F_N(\chi _{n})=6316n-375, \end{aligned}$$
  7. (7)

    The Neighborhood Modified Forgotten index

    $$\begin{aligned} F^*_N(\chi _{n})=4132n-108. \end{aligned}$$
Table 7 Physical properties of drugs used for the treatment of HIV/AIDS patients
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Table 8 Computed topological indices and various drugs
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4 Computed applications of topological descriptors for tenofovir chemical structures

4.1 Regression models

To establish a correlation between the physical properties of various medications utilized in the treatment of HIV/AIDS patients and specific molecular descriptors, we employed a linear regression model. The equation utilized in our study is as follows:

$$\begin{aligned} P_i = a_{0}+a_{1}[TI] \end{aligned}$$

where \(P_i\) is a physical properties, \(a_{0}\) denotes a constant, \(a_{1}\) denotes a regression coefficient, and TI denotes a topological index (Fig. 3).

  1. (1)

    Regression models for Neighborhood first Zagreb index: \(NM_{1}(\chi _{1})\)

    Boiling point = \(227.87+1.0859[ NM_{1}(\chi _{1})]\)

    Enthalpy of vaporization = \(27.57+0.1762[NM_{1}(\chi _{1})]\)

    Flash point = \(91.175+0.657[NM_{1}(\chi _{1})]\)

    Molar refractivity = \(9.9+0.2891[NM_{1}(\chi _{1})]\)

    Molar volume = \(45.7+2.791[ NM_{1}(\chi _{1})]\)

    Polarizability = \(5.45+1.91[ NM_{1}(\chi _{1})]\)

    Surface tension = \(46.9+0.391[ NM_{1}(\chi _{1})]\)

  2. (2)

    Regression models for Neighborhood second Zagreb index: \(NM_{2}(\chi _{1})\)

    Boiling point = \(220.13+0.412[NM_{2}(\chi _{1})]\)

    Enthalpy of vaporization = \(24.15+0.065[NM_{2}(\chi _{1})]\)

    Flash point = \(88.3+0.256[NM_{2}(\chi _{1})]\)

    Molar refractivity = \(7.19+0.112[NM_{2}(\chi _{1})]\)

    Molar volume = \(43.075+2.623[NM_{2}(\chi _{1})]\)

    Polarizability = \(4.65+1.43[NM_{2}(\chi _{1})]\)

    Surface tension = \(45.9+1.631[NM_{2}(\chi _{1})]\)

  3. (3)

    Regression models for Neighborhood Hyper index: \(HM_{N}(\chi _{1})\)

    Boiling point = \(226.69+0.1013[HM_{N}(\chi _{1})]\)

    Enthalpy of vaporization = \(25.69+0.017[HM_{N}(\chi _{1})]\)

    Flash point = \(89.19+0.062[HM_{N}(\chi _{1})]\)

    Molar refractivity = \(17.19+0.025[HM_{N}(\chi _{1})]\)

    Molar volume = \(47.7+0.791[HM_{N}(\chi _{1})]\)

    Polarizability = \(4.45+2.91[HM_{N}(\chi _{1})]\)

    Surface tension = \(47.7+1.91[HM_{N}(\chi _{1})]\)

  4. (4)

    Regression models for Neighborhood redefined \(3^{rd}\) Zagreb index: \(ND_{6}(\chi _{1})\)

    Boiling point = \(217.18+0.036[ND_{6}(\chi _{1})]\)

    Enthalpy of vaporization = \(23.14+0.0061[ND_{6}(\chi _{1})]\)

    Flash point = \(85.44+0.022[ND_{6}(\chi _{1})]\)

    Molar refractivity = \(6.74+0.0095[ND_{6}(\chi _{1})]\)

    Molar volume = \(48.23+3.75[ND_{6}(\chi _{1})]\)

    Polarizability = \(6.32+1.85[ND_{6}(\chi _{1})]\)

    Surface tension = \(46.45+1.36[ND_{6}(\chi _{1})]\)

Table 9 Correlation coefficients
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Table 10 Statistical parameters for the linear QSPR model for \(NM_{1}(\chi )\)
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Table 11 Statistical parameters for the linear QSPR model for \(NM_{2}(\chi )\)
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Table 12 Statistical parameters for the linear QSPR model for \(HM_{N}(\chi )\)
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Table 13 Statistical parameters for the linear QSPR model for \(ND_{6}(\chi )\)
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Table 14 Standard error of estimate
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Table 15 Correlation determination
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Table 16 Comparison of the boiling point calculated and real values using regression models
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Table 17 Comparison of the enthalpy of vaporization calculated and real values using regression models
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Table 18 Comparison of the flash point calculated and real values using regression models
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Table 19 Comparison of the molar refractivity calculated and real values using regression models
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4.2 Standard error of estimate

The standard error of estimate is a measurement of the range of an observation taken near the calculated regression line. The standard error of estimate for the four physical parameters that correlate to each topological index is shown in Table 14.

4.3 Correlation determination

The proportion of correlation which provides more details about the connections among variables, is what the correlation determination explains.

  1. (1)

    Calculation of Topological indices and Comparison with Physical Property Correlation Coefficients. Table 7 displays the physical characteristics of 3 as listed above, medicines that are used to treat HIV/AIDS patient. Table 8 lists the 4 topological indices that were calculated using the graphs made from the pharmacological molecules.

  2. (2)

    The calculated physical characteristics with regard to each topological index are shown in the Tables 9 and 15. The graph illustrates the correlation coefficient and correlation determination between several topological indices and various physical parameters, including boiling point, enthalpy of vaporisation molar volume, flash point, polarizability, surface tention and molar refractivity. The values of eleven degree-based topological indices and seven physical features of medicine are correlated using various statistical parameters shown in Tables 10, 11, 12 and 13. The melting point of HIV/AIDS medications and the degree-based topological index were not found to be correlated.

  3. (3)

    This section draws comparisons between estimated values from our regression models and known values. The comparison of each physical attribute is provided in Tables 16,17, 18 and 19. Examining correlation coefficients horizontally for the physical properties being taken into account, we observe that greatest correlation coefficient \(NM_1(\chi )\) descriptor is given for boiling point \((r_i=0.999)\), enthalpy \((r_i=0.999)\), flash point \((r_i=0.999)\) and molar refractivity \((r_i=0.999)\). Greatest correlation coefficient \(NM_2(\chi )\) descriptor is given for boiling point \((r_i=0.999)\), enthalpy \((r_i=0.999)\), flash point \((r_i=0.999)\) and molar refractivity \((r_i=0.999)\). Greatest correlation coefficient \(HM_{N}(\chi )\) descriptor is given for boiling point \((r_i=0.998)\), enthalpy \((r_i=0.999)\) and flash point \((r_i=0.998)\) and molar refractivity \((r_i=0.932)\). Greatest correlation coefficient \(ND_{6}(\chi )\) descriptor is given for boiling point \((r_i=0.999)\), enthalpy \((r_i=0.999)\), flash point \((r_i=0.998)\) and molar refractivity \((r_i=0.999)\). In a vertical perspective, boiling point also has a strong association with \(NM_1(\chi )\), \(NM_2(\chi )\), \(HM_{N}(\chi )\) and \(ND_{6}(\chi )\). Enthalpy is strongly correlated with \(NM_1(\chi )\), \(NM_2(\chi )\), \(HM_{N}(\chi )\) and \(ND_{6}(\chi )\). Flash point is strongly correlated with \(NM_1(\chi )\), \(NM_2(\chi )\), \(HM_{N}(\chi )\) and \(ND_{6}(\chi )\). Molar refractivity is strongly correlated with \(NM_1(\chi )\), \(NM_2(\chi )\), \(HM_{N}(\chi )\) and \(ND_{6}(\chi )\). According to research, theocratic analysis may enable chemists and others in the pharmaceutical sector to forecast the features of anti-tuberculosis medications without having to do costly experiments. Depending on the range of topological indices that are estimated in this work, it is also possible that alternative formulations of these medications might be employed to treat various disorders. We have discovered the correlation coefficient for several topological indices in this work, which will assist the chemist in creating novel medications based on the combination of positively high correlated ones based on the combination of highly associated positively.

5 Conclusion

In conclusion, this study highlights the significance of topological indices in the field of molecular chemistry and pharmaceuticals. By exploring the topological features of Tenofovir molecular structures, the researcher has computed various degree-based and neighborhood-based indices, and determined their polynomial expressions. Through this analysis, the study provides a deeper understanding of the relationships among the structural features of Tenofovir, which can aid in the development of novel treatments for AIDS and HIV. The findings of this research could be used to forecast various qualities, such as boiling point, molar volume, and surface tension, which are critical in drug creation. Overall, this study demonstrates the value of topological index computation in medical, chemical, pharmaceutical, and biological sciences, and how it can contribute to the development of innovative medications, vaccines, and treatments.