QSPR analysis of some novel neighbourhood degree-based topological descriptors

Abstract

Topological index is a numerical value associated with a chemical constitution for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. In this work, some new indices based on neighborhood degree sum of nodes are proposed. To make the computation of the novel indices convenient, an algorithm is designed. Quantitative structure property relationship (QSPR) study is a good statistical method for investigating drug activity or binding mode for different receptors. QSPR analysis of the newly introduced indices is studied here which reveals their predicting power. A comparative study of the novel indices with some well-known and mostly used indices in structure-property modelling and isomer discrimination is performed. Some mathematical properties of these indices are also discussed here.

Introduction

The graph theory is a significant part of applied mathematics for modeling real life problems. The chemical graph theory, a fascinating branch of graph theory, provides many information on chemical compounds using an important tool called the topological index [4, 43]. Theoretical molecular descriptors alias topological indices are graph invariants that play an important role in chemistry, pharmaceutical sciences, materials science, engineering and so forth. Its role on QSPR/QSAR analysis [2, 22, 23, 37, 38], to model physical and chemical properties of molecules is also remarkable. Among several types of topological indices, vertex degree based [15] topological indices are most investigated and widely used. The first vertex degree based topological index is proposed in 1975 by Randić [36] known as Connectivity index or Randic index. Connectivity index is defined by

$$\begin{aligned} R(G)= \sum \limits _{uv \in E(G)}\frac{1}{\sqrt{d_{G}(u)d_{G}(v)}}, \end{aligned}$$

where $d_{G}(u)$, $d_{G}(v)$ represent the degree of nodes u,v in the vertex set V(G) of a molecular graph G. By molecular graph, we mean a simple connected graph considering atoms of chemical compound as vertices and the chemical bonds between them as edges. E(G) is the edge set of G. The inverse Randic index [19] is given by

$$\begin{aligned} \mathrm{RR} (G)=\sum \limits _{uv \in E(G)}\sqrt{d_{G}(u)d_{G}(v)}. \end{aligned}$$

The Zagreb indices, introduced by Gutman and Trinajestić [20], are defined as follows:

$$\begin{aligned} M_{1}(G)= & {} \sum \limits _{v \in V(G)}d_{G}(v)^{2} =\sum \limits _{uv \in E(G)}[d_{G}(u) + d_{G}(v)], \\ M_{2}(G)= & {} \sum \limits _{uv \in E(G)}[d_{G}(u)d_{G}(v)]. \end{aligned}$$

Furtula et al. [12] have introduced the forgotten topological index as follows:

$$\begin{aligned} F(G) = \sum \limits _{v \in V(G)}d_{G}(v)^{3} = \sum \limits _{uv \in E(G)}[d_{G}(u)^{2} + d_{G}(v)^{2}]. \end{aligned}$$

Zhou and Trinanjstić have designed the sum connectivity index [49] which is as follows:

$$\begin{aligned} \mathrm{SCI}(G)= \sum \limits _{uv \in E(G)}\frac{1}{\sqrt{d_{G}(u)+d_{G}(v)}}. \end{aligned}$$

The symmetric division degree index [44] is defined as

$$\begin{aligned} \mathrm{SDD}(G)= \sum \limits _{uv \in E(G)}[\frac{d_{G}(u)}{d_{G}(v)} + \frac{d_{G}(v)}{d_{G}(u)}]. \end{aligned}$$

The redefined third Zagreb index [39] is defined by

$$\begin{aligned} \mathrm{ReZG}_{3}(G) = \sum \limits _{uv \in E(G)}d_{G}(u)d_{G}(v)[d_{G}(u)+d_{G}(v)]. \end{aligned}$$

For more study about degree based topological indices, readers are referred to the articles [5, 10, 24, 25, 27, 32]. Recently, the present authors introduced some new indices [30, 31] based on neighborhood degree sum of nodes. As a continuation, we present here some new topological indices, named as first NDe index ($\mathrm{ND}_{1}$), second NDe index ($\mathrm{ND}_{2}$), third NDe index ($\mathrm{ND}_{3}$), fourth NDe index ($\mathrm{ND}_{4}$), fifth NDe index ($\mathrm{ND}_{5}$), and sixth NDe index ($\mathrm{ND}_{6}$) and defined as

$$\begin{aligned} \mathrm{ND}_{1}(G)= & {} \sum \limits _{uv \in E(G)}\sqrt{\delta _{G}(u)\delta _{G}(v)} ,\\ \mathrm{ND}_{2}(G)= & {} \sum \limits _{uv \in E(G)}\frac{1}{\sqrt{\delta _{G}(u)+\delta _{G}(v)}} ,\\ \mathrm{ND}_{3}(G)= & {} \sum \limits _{uv \in E(G)}\delta _{G}(u)\delta _{G}(v)[\delta _{G}(u)+\delta _{G}(v)] ,\\ \mathrm{ND}_{4}(G)= & {} \sum \limits _{uv \in E(G)}\frac{1}{\sqrt{\delta _{G}(u)\delta _{G}(v)}} ,\\ \mathrm{ND}_{5}(G)= & {} \sum \limits _{uv \in E(G)} \left[ \frac{\delta _{G}(u)}{\delta _{G}(v)} + \frac{\delta _{G}(v)}{\delta _{G}(u)}\right] ,\\ \mathrm{ND}_{6}(G)= & {} \sum \limits _{uv \in E(G)}[d_{G}(u)\delta _{G}(u)+d_{G}(v)\delta _{G}(v)] , \end{aligned}$$

where $\delta _{G}(u)$ is the sum of degrees of all neighboring vertices of $u \in V(G)$, i.e, $\delta _{G}(u)=\sum \nolimits _{v \in N_{G}(u)}d_{G}(v)$, $N_{G}(u)$ being the set of adjacent vertices of u. The goal of this article is to check the chemical applicability of the above newly designed indices and discuss about some bounds of them in terms of other topological descriptors to visualize the indices mathematically.

Table 1 Experimental values of physico-chemical properties for octane isomers

We construct the results into two different parts. We start the first part with an algorithm for computing the indices and then some statistical regression analysis have been made to check the efficiency of the novel indices to model physical and chemical properties. Then, we would like to test their degeneracy. It follows a comparative study of these indices with other topological indices. This part ends with a discussion about the applications of the present work. The second part deals with some mathematical relation of these indices with some other well-known indices.

Computational aspects

In this section, we have designed an algorithm to make the computation of the novel indices convenient.

To make it simple and understandable, we have considered some variables and matrices. We have used conn [E][2] matrix to store the connection details among vertices, whereas deg [V][2] and $\delta [V][2]$ is the matrix to store degree of each vertex and neighborhood degree sum of vertex respectively. The novel indices can be considered as function of $\delta _{G}(u)$,$\delta _{G}(v)$,$d_{G}(u)$, and $d_{G}(v)$ i.e., f($\delta _{G}(u)$,$\delta _{G}(v)$,$d_{G}(u)$, $d_{G}(v)$).

Newly introduced indices in QSPR analysis

In this section, we have studied about the newly designed topological indices to model physico-chemical properties [Acentric Factor (Acent Fac.), entropy (S), enthalpy of vaporization (HVAP), standard enthalpy of vaporization (DHVAP), and heat capacity at P constant (CP)] of the octane isomers and physical properties [boiling points (bp), molar volumes (mv) at 20$^\circ$C, molar refraction (mr) at 20 $^\circ$C, heats of vaporization (hv) at 25 $^\circ$C, critical temperature (ct), critical pressure (cp) surface tensions (st) at 20 $^\circ$C and melting points (mp)] of the 67 alkanes from n-butanes to nonanes. The experimental values of physico-chemical properties of octane isomers (Table 1) are taken from http://www.moleculardescriptors.eu. The data related to 67 alkanes (Table 9) are compiled from [27]. For comparative study, different well-known existing descriptors are collected form http://www.moleculardescriptors.eu/books/books.htm. First, we have considered the octane isomers (Table 2) and then the 67 alkanes are taken into account.

Table 2 Topological indices of octane isomers

Regression model for octane isomers

We have tested the following linear regression models

$$\begin{aligned} P = m(\mathrm{TI})+c, \end{aligned}$$

(1)

where P is the physical property and TI is the topological index. Using the above formula, we have the following linear regression models for different neighborhood degree sum based topological indices.

Fig. 1

Correlation of $\mathrm{ND}_{1}$ index with S, Acent Fac., and DHVAP for octane isomers

1. $\mathrm{ND}_{1}$ index:

$$\begin{aligned} S&=141.1521-[\mathrm{ND}_{1} (G)]1.1926\\ \mathrm{{Acent~Fac.}}&=0.627 - [\mathrm{ND}_{1} (G)]0.0097\\ \mathrm{DHVAP}&=11.8017 - [\mathrm{ND}_1 (G)]0.0893 \end{aligned}$$

2. $\mathrm{ND}_{2}$ index:

$$\begin{aligned} S&=39.6776+[\mathrm{ND}_2 (G)]27.1579\\ \mathrm{Acent ~Fac.}&=-0.2058 + [\mathrm{ND}_2 (G)]0.2238\\ \mathrm{DHVAP}&=1.1069 + [\mathrm{ND}_2 (G)]2.0737 \end{aligned}$$

3. $\mathrm{ND}_{3}$ index:

$$\begin{aligned} S&=117.2259-[\mathrm{ND}_3 (G)]0.0088\\ \mathrm{Acent~ Fac.}&= 0.4322 - [\mathrm{ND}_3 (G)]7.2 \times 10^{-5}\\ \mathrm{DHVAP}&=9.9568 - [\mathrm{ND}_3 (G)]0.0006 \end{aligned}$$

4. $\mathrm{ND}_{4}$ index:

$$\begin{aligned} S&=69.8183+[\mathrm{ND}_4 (G)]20.3149\\ \mathrm{Acent~ Fac.}&= 0.04656+ [\mathrm{ND}_4 (G)]0.1651\\ \mathrm{DHVAP}&= 6.2868 + [\mathrm{ND}_4 (G)]1.6206\\ \mathrm{HVAP}&= 55.1172 + [\mathrm{ND}_4 (G)]8.0164 \end{aligned}$$

5. $\mathrm{ND}_{5}$ index:

$$\begin{aligned} S&=144.7836-[\mathrm{ND}_5 (G)]2.5286\\ \mathrm{Acent ~Fac.}&= 0.7245 - [\mathrm{ND}_5 (G)]0.0249\\ \mathrm{DHVAP}&= 10.6958 - [\mathrm{ND}_5 (G)]0.1008\\ \mathrm{CP}&=3.8987+[\mathrm{ND}_5 (G)]1.4447 \end{aligned}$$

6. $\mathrm{ND}_{6}$ index:

$$\begin{aligned} S&=122.3482-[\mathrm{ND}_6 (G)]0.1122\\ \mathrm{Acent~ Fac.}&= 0.4730 -[\mathrm{ND}_6 (G)]0.0009\\ \mathrm{DHVAP}&= 10.3438 - [\mathrm{ND}_6 (G)]0.0081 \end{aligned}$$

Fig. 2

Correlation of $\mathrm{ND}_{2}$ index with S, Acent Fac., and DHVAP for octane isomers

Fig. 3

Correlation of $\mathrm{ND}_{3}$ index with S, Acent Fac., and DHVAP for octane isomers

Fig. 4

Correlation of $\mathrm{ND}_{4}$ index with S, Acent Fac., and DHVAP for octane isomers

Fig. 5

Correlation of $\mathrm{ND}_{5}$ index with S, Acent Fac., and DHVAP for octane isomers

Fig. 6

Correlation of $\mathrm{ND}_{6}$ index with S, Acent Fac., and DHVAP for octane isomers

The correlations of the novel descriptors with different physico-chemical properties are depicted in the Figs. 1, 2, 3, 4, 5 and 6.

Now we describe above linear models in the Tables 3, 4, 5, 6, 7 and 8. Here c, m, r, SE, F, SF stands for intercept, slope, correlation coefficient, standard error, F test, and significance F respectively. Correlation coefficient tells how strong the linear relationship is. The standard error of the regression is the precision that the regression coefficient is measured. To check whether the results are reliable, Significance F can be useful. If this value is less than 0.05, then the model is statistically significant. If significance F is greater than 0.05, it is probably better to stop using that set of independent variable.

Table 3 Statical parameters linear QSPR model for $\mathrm{ND}_{1}(G)$

Table 4 Statical parameters of linear QSPR model for $\mathrm{ND}_2 (G)$

Table 5 Statical parameters of linear QSPR model for $\mathrm{ND}_3 (G)$

Table 6 Statical parameters of linear QSPR model for $\mathrm{ND}_4 (G)$

Table 7 Statical parameters of linear QSPR model for $\mathrm{ND}_5 (G)$

Table 8 Statical parameters of linear QSPR model for $\mathrm{ND}_6 (G)$

Regression model for 67 alkanes

We have tested here the model described in (1) for 67 alkanes from n-butanes to nonanes. we have the following linear regression models for different neighborhood degree sum-based topological indices.

1. $\mathrm{ND}_{1}$ index:

$$\begin{aligned} \mathrm{bp}&=-8.8069+[\mathrm{ND}_1 (G)]4.0114\\ \mathrm{ct}&=135.4475+[\mathrm{ND}_1 (G)]5.1317\\ \mathrm{cp}&=34.6560-[\mathrm{ND}_1 (G)]0.2721\\ \mathrm{mv}&=100.8619+[\mathrm{ND}_1 (G)]2.0398\\ \mathrm{mr}&=20.1480+[\mathrm{ND}_1 (G)]0.6398\\ \mathrm{hv}&=21.8703+[\mathrm{ND}_1 (G)]0.5616\\ \mathrm{st}&=14.3557+[\mathrm{ND}_1 (G)]0.2177\\ \mathrm{mp}&=-131.654+[\mathrm{ND}_1 (G)]0.7933 \end{aligned}$$

2. $\mathrm{ND}_{2}$ index:

$$\begin{aligned} \mathrm{bp}&=-95.9303+[\mathrm{ND}_2 (G)]84.2834\\ \mathrm{ct}&=47.6081+[\mathrm{ND}_2 (G)]98.1177\\ \mathrm{cp}&=43.8159-[\mathrm{ND}_2 (G)]7.0529\\ \mathrm{mv}&=50.7515+[\mathrm{ND}_2 (G)]45.1187\\ \mathrm{mr}&=6.7917+[\mathrm{ND}_2 (G)]13.1956\\ \mathrm{hv}&=3.9821+[\mathrm{ND}_2 (G)]14.0794\\ \mathrm{st}&=9.6530+[\mathrm{ND}_2 (G)]4.5432\\ \mathrm{mp}&=-150.6221+[\mathrm{ND}_2 (G)]17.6077 \end{aligned}$$

3. $\mathrm{ND}_{3}$ index:

$$\begin{aligned} \mathrm{bp}&=60.7738+[\mathrm{ND}_3 (G)]0.0372\\ \mathrm{ct}&=220.3722+[\mathrm{ND}_3 (G)]0.0508\\ \mathrm{cp}&=29.0689-[\mathrm{ND}_3 (G)]0.0019\\ \mathrm{mv}&=140.8691+[\mathrm{ND}_3 (G)]0.0159\\ \mathrm{mr}&=32.2248+[\mathrm{ND}_3 (G)]0.0054\\ \mathrm{hv}&=32.9914+[\mathrm{ND}_3 (G)]0.0043\\ \mathrm{st}&=18.2408+[\mathrm{ND}_3 (G)]0.0020\\ \mathrm{mp}&=-118.9098+[\mathrm{ND}_3 (G)]0.0078 \end{aligned}$$

Table 9 Experimental values of physical properties for 67 alkanes

Table 10 Topological indices for 67 alkanes

4. $\mathrm{ND}_{4}$ index:

$$\begin{aligned} \mathrm{bp}&=-69.6148+[\mathrm{ND}_4 (G)]100.5729\\ \mathrm{ct}&=86.3449+[\mathrm{ND}_4 (G)]112.5233\\ \mathrm{cp}&=42.1443-[\mathrm{ND}_4 (G)]8.7145\\ \mathrm{mv}&=75.6867+[\mathrm{ND}_4 (G)]48.0647\\ \mathrm{mr}&=14.6003+[\mathrm{ND}_4 (G)]13.7704\\ \mathrm{hv}&=9.3610+[\mathrm{ND}_4 (G)]16.3338\\ \mathrm{st}&=12.1161+[\mathrm{ND}_4 (G)]4.8742\\ \mathrm{mp}&=-146.0101+[\mathrm{ND}_4 (G)]21.4463 \end{aligned}$$

5. $\mathrm{ND}_{5}$ index:

$$\begin{aligned} \mathrm{bp}&=-62.0831+[\mathrm{ND}_5 (G)]11.0891\\ \mathrm{ct}&=73.9689+[\mathrm{ND}_5 (G)]13.7538\\ \mathrm{cp}&=38.1279-[\mathrm{ND}_5 (G)]0.7430\\ \mathrm{mv}&=74.7263+[\mathrm{ND}_5 (G)]5.5594\\ \mathrm{mr}&=12.0902+[\mathrm{ND}_5 (G)]1.7348\\ \mathrm{hv}&=11.8876+[\mathrm{ND}_5 (G)]1.7079\\ \mathrm{st}&=10.2382+[\mathrm{ND}_5 (G)]0.6757\\ \mathrm{mp}&=-142.0736+[\mathrm{ND}_5 (G)]2.2172 \end{aligned}$$

6. $\mathrm{ND}_{6}$ index:

$$\begin{aligned} \mathrm{bp}&=35.3191+[\mathrm{ND}_6 (G)]0.5065\\ \mathrm{ct}&=187.4769+[\mathrm{ND}_6 (G)]0.6782\\ \mathrm{cp}&=30.8960-[\mathrm{ND}_6 (G)]0.0291\\ \mathrm{mv}&=127.7664+[\mathrm{ND}_6 (G)]0.2304\\ \mathrm{mr}&=28.1203+[\mathrm{ND}_6 (G)]0.0754\\ \mathrm{hv}&=29.6437+[\mathrm{ND}_6 (G)]0.0609\\ \mathrm{st}&=16.9413+[\mathrm{ND}_6 (G)]0.0266\\ \mathrm{mp}&=-123.8778+[\mathrm{ND}_6 (G)]0.1045 \end{aligned}$$

The statistical parameters like previous discussion are used in Tables 11, 12, 13, 14, 15 and 16 to interpret the above regression models, where N denotes the total number of alkanes under consideration.

Table 11 Statical parameters of linear QSPR model for $\mathrm{ND}_{1} (G)$

Table 12 Statical parameters of linear QSPR model for $\mathrm{ND}_{2}(G)$

Table 13 Statical parameters of linear QSPR model for $\mathrm{ND}_{3}$(G)

Table 14 Statical parameters of linear QSPR model for $\mathrm{ND}_{4}(G)$

Table 15 Statical parameters of linear QSPR model for $\mathrm{ND}_{5}(G)$

Table 16 Statistical parameters of linear QSPR model for $\mathrm{ND}_{6}(G)$

Several interesting observations on the data presented in Table 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16 can be made. From Table 3, the correlation coefficient of $\mathrm{ND}_1$ index with entropy, acentric factor and DHVAP for octane isomers are found to be good (Fig. 1). Specially, it is strongly correlated with acentric factor having correlation coefficient $r=-0.9904$. Also, the correlation of this index is good for the physical properties of 67 alkanes except for cp and mp having correlation coefficient values − 0.6941 and 0.2516, respectively. The range of correlation coefficient values lies from 0.7436 to 0.8981.

The QSPR analysis of $\mathrm{ND}_2$ index reveals that this index is suitable to predict entropy, acentric factor and DHVAP of octane isomers (Fig. 2). In addition, one can say from Table 12 that, this index have remarkably good correlations with the physical properties of alkanes except mp. The correlation coefficients lies from 0.809 to 0.9638 except mp ($r=0.2862$). Surprisingly, the correlation of $\mathrm{ND}_2$ with hv is very high with correlation coefficient value 0.9638.

Table 13 shows that $\mathrm{ND}_3$ index is inadequate for any structure property correlation in the case of alkanes having the correlation coefficient values from 0.2036 to 0.7318. But, from Table 5, we can see that $\mathrm{ND}_3$ is well correlated with entropy and acentric factor with correlation coefficients $-0.9387$ and $-0.9765$ respectively.

The QSPR analysis of $\mathrm{ND}_4$ index shows that this index is well correlated with entropy, acentric factor, DHVAP, and HVAP for octane isomers (Table 6). Table 14 shows that $\mathrm{ND}_4$ index is inadequate for structure property correlation in case of alkanes except cp and hv having correlation coefficients -0.8634 and 0.8679, respectively.

From Table 7, one can say that $\mathrm{ND}_5$ does not sound so good except CP having correlation coefficient 0.8478. But this index can be considered as an useful tool to predict the physical properties of alkanes except cp and mp. This index is suitable to model bp, ct, mv, mr, hv, st with correlation coefficients 0.9166, 0.9779, 0.8881, 0.9319, 0.8950, and 0.9267, respectively.

The QSPR analysis of $\mathrm{ND}_6$ index reveals that the correlation coefficient of this index with the physical properties of alkanes are very poor (Table 16). The range of correlation coefficient values lies from 0.2192 to 0.7823. But, when we look into the Table 8, we can say that this index has the ability to model entropy, acentric factor, and DHVAP for octane isomers.

Now we compare the modelling ability of the novel indices with some well-known and mostly used indices that include: First Zagreb index ($M_1$), second Zagreb index ($M_2$), Forgotten topological index (F), Sum connectivity index (SCI), Randić index (R), symmetric division deg index (SDD), Weiner index (W) [47], hyper Weiner index [37] (WW), terminal Wiener index (TW) [18], Schultz index (Sc) [41], first ($S_1$) and second status connectivity index ($S_2$) [34], Gutman index (GI) [16], degree distance index (DD) [6], inverse sum indeg status (ISIS) index [7], total eccentricity connectivity index (TECI) [1], first Zagreb eccentricity connectivity Index (ZECI$_1$) [13], first ($\xi _1$) and second eccentricity connectivity index ($\xi _2$) [29, 46], connective eccentricity index (CEI) [29, 46], vertex adjacency energy (E) [17], Laplacian energy (LE) [21], atom bond connectivity index (ABC) [8], augmented Zagreb index (AZI) [11], geometric arithmetic index (GA) [45], harmonic index (H) [9], Ashwini Index (A) [33], SM-index (SM) [42], vertex Zagreb energy ($Z_{1}E$) [26], forgotten energy (FE) [26], harmonic energy (HE) [26], geometric-arithmetic energy (GAE) [40], degree-sum energy (DSE) [35], sum-connectivity energy (SCE) [48], and Randić energy (RE) [3]. From Tables 3, 4, 5, 6, 7 to 8, it is clear that among six new indices, the $\mathrm{ND}_1$ index can model acentric factor of octane isomers with excellent accuracy. To investigate the predictability of different well-established descriptors for acentric factor of octanes, linear regression analysis is performed and the outcomes are reported in Tables 17, 18, 19 and 20. From those findings, several observations can be made. The modulus of the correlation coefficient and the F value of $\mathrm{ND}_1$ index is significantly high compared to the existing indices listed in Tables 17, table:comp2, table:comp3 and 20. The standard error and the SF value of the $\mathrm{ND}_1$ index is lower than that of the indices reported in Tables 17, 18, 19 and 20. Thus, it can be concluded that the $\mathrm{ND}_1$ index is efficient in predicting acentric factor of octanes with high accuracy compared to several well-known and mostly utilised molecular descriptors.

Table 17 Statical parameters of linear QSPR model of acentric factor for some degree based indices

Table 18 Statical parameters of linear QSPR model of acentric factor for some distance based indices

Table 19 Statical parameters of linear QSPR model of acentric factor for some eccentricity based indices

Table 20 Statical parameters of linear QSPR model of acentric factor for some graph energies

Now we are going to compare the new descriptors with the existing descriptors in structure-property modelling for 67 alkanes. Statistical parameters of linear regression models of different degree based indices, distance based indices and spectral indices are reported in [24, 26, 42]. We listed the correlation coefficients of those models in Tables 21, 22 and 23 for critical temperature (ct), critical pressure (cp) and surface tension (st). Form Tables 11, 12, 13, 14, 1516, 21, 22, 23, one can draw the following observations. Among all the newly proposed indices and the already existing indices listed in Tables 21, 22 and 23, the $\mathrm{ND}_5$ index has remarkable correlation with ct and st, whereas for cp, the $\mathrm{ND}_2$ index sounds the best. The rest parameters [24, 26, 42] are also in favour of $\mathrm{ND}_2$ and $\mathrm{ND}_5$ indices. Therefore, we can conclude that $\mathrm{ND}_2$ and $\mathrm{ND}_5$ indices outperform several well-established and mostly utilised descriptors in modelling cp, ct, and st for alkanes.

Table 21 Correlation coefficients of some degree based indices with different physical properties for 67 alkanes

Table 22 Correlation coefficients of some distance based indices with different physical properties for 67 alkanes

Table 23 Correlation coefficients of some graph energies with different physical properties for 67 alkanes

Correlation with some well-known indices

In this section, we investigate the correlation between the new indices and some well-known indices for octane isomers. It is clear from Table 24, that the new indices have a high correlation with the well-established indices except $\mathrm{ND}_5$ index. Highest correlation coefficient ($r=0.9977$) is between $\mathrm{ND}_1$ and $M_{2}$. From Table 25, one can say that $\mathrm{ND}_5$ has significantly low correlation coefficient with other indices. So we can conclude that $\mathrm{ND}_5$ is independent among five indices. A correlation graph (Fig. 7) is drawn considering indices as vertices and two vertices are adjacent if and only if $\vert r \vert \ge 0.95$.

Fig. 7

Correlation graph of novel indices with some well-known indices for decane isomer

Table 24 Correlation with some well-known indices

Table 25 Correlation among new indices

Degeneracy

The objective of a topological index is to encipher the structural property as much as possible. Different structural formulae should be distinguished by a good topological descriptor. A major drawback of most topological indices is their degeneracy, i.e., two or more isomers possess the same topological index. Topological indices having high discriminating power captures more structural information. We use the measure of degeneracy known as sensitivity introduced by Konstantinova [28], which is defined as follows:

$$\begin{aligned} S_{I}=\frac{N-N_{I}}{N}, \end{aligned}$$

where N is the total number of isomers considered and $N_I$ is the number of them that cannot be distinguished by the topological index I. As $S_I$ increases, the isomer-discrimination power of topological indices increases. The vertex degree-based topological indices have more discriminating power in comparison with other classes of molecular descriptors. For octane and decane isomers, the newly introduced indices exhibit better response compared to some well-known degree-based indices (Table 26).

Table 26 Measure of sensitivity ($S_I$) of different indices for octane and decane isomers

Applications

QSPR analysis is a powerful investigation for breaking down a molecule into a series of numerical values describing its relevant physico-chemical properties and biological activities. Descriptors having the strongest correlation in this study give information about essential functional groups of compounds under consideration. Accordingly, we can regulate pharmacological action or physico-chemical properties of drugs by modifying certain groups in the structure of medications. It is usually very costly to test a compound using a wet lab, but the QSPR study allow that cost to be reduced. This is generally used to analyze biological activities with specific properties associated with the structures and is helpful in understanding how molecular attributes in a drug effect biological activities. QSPR approaches can be used to develop models which can predict properties or activities of organic chemical. An efficient way of encoding structures with determined topological index is, therefore, necessary for the construction of accurate models. The indices used for the creation of model can offer a chance to concentrate on particular characteristics that account for the activity or property of interest in the compounds. QSPR analysis of some newly designed indices using octane isomers and alkanes is performed in this work. It has been shown that these indices can be considered as useful molecular descriptors in QSPR research. They yield excellent correlation with S, Acent Fac, HVAP, DHVAP, CP for octane isomers and bp, ct, cp, mv, mr,hv, st, mp for alkanes. Their isomer discrimination ability is also remarkable for octane and decane isomers. These indices are an extension of some well-known degree-based topological indices namely R, $\mathrm{SCI}$, $\mathrm{SDD}$, and $\mathrm{ReZG}_{3}$. Sometimes the predictive power of these indices is superior, sometimes little bit inferior than that of the old indices. But the degeneracy test on Table 26, assures the supremacy of newly designed indices in comparison to the old indices. It is worth discussing the mathematical properties of the novel descriptors discussed in the following section.

Mathematical properties

In this section, we discuss about some bounds of the newly proposed indices with some well-known indices. Throughout this section, we consider simple connected graph. We construct this section with some standard inequalities. We start with the following inequality.

Lemma 1

(Radon’s inequality) If $x_{i}, y_{i}> 0, i=1,2, \ldots ,n, t>0$, then

$$\begin{aligned} \frac{\sum \nolimits _{i = 1}^{n}x_{i}^{t+1}}{\sum \nolimits _{i = 1}^{n}y_{i}^{t}} \ge \frac{\left( \sum \nolimits _{i = 1}^{n}x_{i}\right) ^{t+1}}{\left( \sum \nolimits _{i = 1}^{n}y_{i}\right) ^{t}}, \end{aligned}$$

(2)

where equality holds iff $x_i=ky_i$ for some constant k, $\forall i=1,2, \ldots ,n.$

Proposition 1

For a graph G having m edges with neighborhood version of second Zagreb index $M_2^{*} (G)$ [31], we have

$$\begin{aligned} \mathrm{ND}_{1}(G) \le \sqrt{mM_2^{*} (G)}, \end{aligned}$$

(3)

where equality holds iff G is regular or complete bipartite graph.

Proof

For a graph G, $M_2^{*} (G)= \sum \nolimits _{uv \in E(G)}\delta _{G}(u)\delta _{G}(v)$. Now considering $x_{i} =1, y_{i} =\delta _{G}(u)\delta _{G}(v), t = \frac{1}{2}$, in (2), we obtain

$$\begin{aligned} \frac{\sum \nolimits _{uv \in E(G)}1}{\sum \nolimits _{uv \in E(G)}(\delta _{G}(u)\delta _{G}(v))^{\frac{1}{2}}} \ge \frac{ \left( \sum \nolimits _{uv \in E(G)}1 \right) ^{\frac{3}{2}}}{ \left( \sum \nolimits _{uv \in E(G)}\delta _{G}(u)\delta _{G}(v)\right) ^{\frac{1}{2}}}.\nonumber \\ \end{aligned}$$

(4)

Now using the definition of $\mathrm{ND}_{1}$ and $M_2^{*}$ indices, we can easily obtain the required bound (3). Equality in (4) holds iff $\delta _{G}(u)\delta _{G}(v)=k$, a constant $\forall uv \in E(G)$. So the equality in (3) holds iff G is regular or complete bipartite graph.

Lemma 2

Let $\mathbf {x}=(x_1,x_2, \ldots ,x_n)$ and $\mathbf {y}=(y_1,y_2, \ldots ,y_n)$ be sequence of real numbers. Also let $\mathbf {z}=(z_1,z_2, \ldots ,z_n)$ and $\mathbf {w}=(w_1,w_2, \ldots ,w_n)$ be non-negative sequences. Then

$$\begin{aligned} \sum \limits _{i = 1}^{n}w_{i}\sum \limits _{i = 1}^{n}Z_{i}x_{i}^{2}+\sum \limits _{i = 1}^{n}z_{i}\sum \limits _{i = 1}^{n}w_{i}y_{i}^{2} \ge 2 \sum \limits _{i = 1}^{n}z_{i}x_{i}\sum \limits _{i = 1}^{n}w_{i}y_{i},\nonumber \\ \end{aligned}$$

(5)

In particular, if $z_i$ and $w_i$ are positive, then the equality holds iff $\mathbf {x}=\mathbf {y}=\mathbf {k}$, where $\mathbf {k}=(k,k,\ldots ,k)$, a constant sequence.

Proposition 2

For a graph G having m edges with neighbourhood version of second Zagreb index $M_2^{*}(G)$, we have

$$\begin{aligned} \mathrm{ND}_{1}(G) \le \frac{(m+M_2^{*}(G))}{2}, \end{aligned}$$

(6)

where equality holds iff G is $P_2$.

Proof

Considering $x_{i} =\delta _{G}(u)\delta _{G}(v), y_{i} =1, z_{i} =1, w_{i} =1$, in (5), we get

$$\begin{aligned}&\sum \limits _{uv \in E(G)}1\sum \limits _{uv \in E(G)}\delta _{G}(u)\delta _{G}(v) + \sum \limits _{uv \in E(G)}1\sum \limits _{uv \in E(G)}1\\&\quad \ge 2 \sum \limits _{uv \in E(G)}\sqrt{\delta _{G}(u)\delta _{G}(v)}\sum \limits _{uv \in E(G)}1. \end{aligned}$$

After using the definition of $\mathrm{ND}_1$ and $M_2^{*}$ indices we can obtain

$$\begin{aligned} m M_2^{*}(G)+m^{2} \ge 2m\mathrm{ND}_1(G). \end{aligned}$$

After simplification, the required bound is obvious.

From Lemma 2, the equality in (6) holds iff $\delta _{G}(u)\delta _{G}(v)=1 \forall uv \in E(G)$,i.e. G is $P_2$.

Remark

By arithmetic mean $\ge$ geometric mean, we can write

$$\begin{aligned} \frac{(m+M_2^{*}(G))}{2} \ge \sqrt{m M_{2}^{*}(G)}. \end{aligned}$$

So the upper bound of $\mathrm{ND}_1 (G)$ obtained in Proposition 1, is better than that obtained in Proposition 2.

Proposition 3

For a graph G having second Zagreb index $M_2 (G)$, forgotten topological index F(G) , neighbourhood version of hyper Zagreb index $HM_N (G)$ [31], neighbourhood Zagreb index $M_N (G)$ [30], we have

$$\begin{aligned} \mathrm{ND}_{6}(G) \le \frac{F(G)}{2} + M_{2}(G)+\frac{HM_{N}(G)}{2}-M_{N}(G), \end{aligned}$$

(7)

equality holds iff G is $P_{2}$.

Proof

For a graph G, we have $M_{N}(G) =\sum \nolimits _{v \in V(G)}\delta _{G}(v)^{2} =\sum \nolimits _{uv \in E(G)}[\delta _{G}(u)d_{G}(v)+\delta _{G}(v)d_{G}(u)]$, $HM_{N}(G) = \sum \nolimits _{uv \in E(G)}[\delta _{G}(u)+\delta _{G}(v)]^{2}$. We know that for any two non-negative numbers x, y, arithmetic mean $\ge$ geometric mean, i.e., $\frac{x+y}{2} \ge \sqrt{xy},$ equality holds iff $x=y$. Now considering $x=d_G (u)+d_G (v)$, $y=\delta _G (u)+\delta _G (v)$, we get

$$\begin{aligned}&\frac{[d_G (u)+d_G (v)+\delta _G (u)+\delta _G (v)]}{2} \\&\quad \ge \sqrt{(d_G (u)+d_G (v))(\delta _G (u)+\delta _G (v))}, \end{aligned}$$

squiring both sides, we have

$$\begin{aligned}&4(d_G (u)+d_G (v))(\delta _G (u)+\delta _G (v)) \le [d_G (u)+d_G (v)\\&\quad +\delta _G (u)+\delta _G (v)]^{2}, \end{aligned}$$

which gives

$$\begin{aligned}&2\sum \limits _{uv \in E(G)}[(d_{G}(u)\delta _{G}(u) + d_{G}(v)\delta _{G}(v))(d_{G}(u)\delta _{G}(v)\\&\qquad + d_{G}(v)\delta _{G}(u))] \le \sum \limits _{uv \in E(G)}[d_{G}(u)^{2}+d_{G}(v)^{2}]\\&\qquad +2\sum \limits _{uv \in E(G)}d_{G}(u)d_{G}(v)+ \sum \limits _{uv \in E(G)}[\delta _{G}(u)^{2}+\delta _{G}(v)^{2}]\\&\qquad + 2\sum \limits _{uv \in E(G)}\delta _{G}(u)\delta _{G}(v). \end{aligned}$$

After simplifying and using the formulation of $\mathrm{ND}_6$, F, $M_2$, $\mathrm{HM}_N$, and $M_N$ indices, the required bound is clear. The equality in (7) occurs iff $d_G (u)+d_G (v)= \delta _G (u)+\delta _G (v)$, i.e., G is $P_2$. Hence the proof. $\square$

For a graph G consider

$$\begin{aligned} \Delta _{N}&= \max \lbrace \delta _{G}(v) : v \in V(G) \rbrace ,\\ \delta _{N}&= \min \lbrace \delta _{G}(v) : v \in V(G) \rbrace . \end{aligned}$$

Thus $\delta _{N} \le \delta _{G}(u) \le \Delta _{N}$ for all $u \in V(G)$. Equality holds iff G is regular or complete bipartite graph. Clearly we have the following proposition.

Proposition 4

For a graph G with m number of edges, we have the following bounds.

• (i) $m\delta _{N} \le \mathrm{ND}_{1}(G) \le m\Delta _{N}$,

• (ii) $\frac{m}{\sqrt{2\Delta _{N}}} \le \mathrm{ND}_{2}(G) \le \frac{m}{\sqrt{2\delta _{N}}}$,

• (iii) $2m\delta _{N}^{3} \le \mathrm{ND}_{3}(G) \le 2m\Delta _{N}^{3}$,

• (iv) $\frac{m}{\Delta _{N}} \le \mathrm{ND}_{4}(G) \le \frac{m}{\delta _{N}}$,

• (v) $\frac{F_{N}^{*}(G)-2M_{2}^{*}(G)}{\delta _{N}^{2}}+2m \le \frac{F_{N}^{*}(G)-2M_{2}^{*}(G)}{\Delta _{N}^{2}}+2m$,

where [31] $F_{N}^{*}(G)= \sum \nolimits _{uv \in E(G)}[d_{G}(u)^{2}+d_{G}(v)^{2}].$

Equality holds in each case iff G is regular or complete bipartite graph.

Lemma 3

Let $a_i$ and $b_i$ be two sequences of real numbers with $a_i \ne 0$ ($i=1,2, \ldots ,n$) and such that $pa_i \le b_i \le Pa_i$. Then

$$\begin{aligned} \sum \limits _{i=1}^{n}b_{i}^{2}+pP\sum \limits _{i=1}^{n}a_{i}^{2} \le (P+p)\sum \limits _{i=1}^{n}a_{i}b_{i}. \end{aligned}$$

(8)

Equality holds iff either $b_i=pa_i$ or $b_i=Pa_i$ for every $i=1,2, \ldots ,n.$

Proposition 5

For a graph G with m edges having neighbourhood version of second Zagreb index $M_2^{*}(G)$, we have

$$\begin{aligned} \mathrm{ND}_{1}(G) \ge \frac{M_{2}^{*}(G)+m\delta _{N}\Delta _{N}}{\delta _{N}+\Delta _{N}} \end{aligned}$$

Equality holds iff G is regular or complete bipartite graph.

Proof

Putting $a_{i}=1$, $b_{i}=\sqrt{\delta _{G}(u)\delta _{G}(v)}$, $p=\delta _{N}$, $P=\Delta _{N}$ in 8, we get

$$\begin{aligned}&\sum \limits _{uv \in E(G)}\delta _{G}(u)\delta _{G}(v)+\delta _{N}\Delta _{N}\sum \limits _{uv \in E(G)}1 \\&\quad \le (\delta _{N}+\Delta _{N})\sum \limits _{uv \in E(G)}\sqrt{\delta _{G}(u)\delta _{G}(v)}. \end{aligned}$$

Now applying the definition of $M_2^{*}(G)$, $\mathrm{ND}_1(G)$ in the above inequation, we obtain

$$\begin{aligned} M_{2}^{*}(G) + m\delta _{N}\Delta _{N} \le (\delta _{N}+\Delta _{N}) \mathrm{ND}_{1}(G). \end{aligned}$$

Which implies

$$\begin{aligned} \mathrm{ND}_{1}(G) \ge \frac{M_2^{*}(G)+m\delta _{N}\Delta _{N}}{\delta _{N}+\Delta _{N}}. \end{aligned}$$

Equality holds iff $\sqrt{\delta _{G}(u)\delta _{G}(v)} = \delta _{N}$ or $\sqrt{\delta _{G}(u)\delta _{G}(v)} = \Delta _{N}$ for all $uv \in E(G)$, i.e. G is regular or complete bipartite graph. Hence the proof. $\square$

Proposition 6

For a graph G of size m with fifth version of geometric arithmetic index $GA_{5}$, and second Zagreb index $M_{2}(G)$, we have

• (i) $\mathrm{ND}_{5}(G) \ge \frac{2m^{2}}{ GA_{5}}$,

• (ii) $\mathrm{ND}_{5}(G) \ge \frac{4M_{2}(G)^{2}}{m\Delta _{N}^{2}}-2m$.

Equality in both cases hold iff G is regular or complete bipartite graph.

Proof

• (i) For a graph G, we know that [14] $GA_{5}(G)=\sum \nolimits _{uv \in E(G)}\frac{2\sqrt{\delta _{G}(u)\delta _{G}(v)}}{\delta _{G}(u)+\delta _{G}(v)}$. Now by Cauchy–Schwarz inequality, we have

  $$\begin{aligned}&\left( \sum \limits _{uv \in E(G)}1 \right) ^{2} = \bigg (\sum \limits _{uv \in E(G)}\sqrt{\frac{\delta _{G}(u)+\delta _{G}(v)}{\sqrt{\delta _{G}(u)\delta _{G}(v)}}} \\&\qquad \times \frac{1}{\sqrt{\frac{\delta _{G}(u)+\delta _{G}(v)}{\sqrt{\delta _{G}(u)\delta _{G}(v)}}}}\bigg )^{2} \\&\quad \le \sum \limits _{uv \in E(G)}\frac{\delta _{G}(u)+\delta _{G}(v)}{\sqrt{\delta _{G}(u)\delta _{G}(v)}}\sum \limits _{uv \in E(G)}\frac{\sqrt{\delta _{G}(u)\delta _{G}(v)}}{\delta _{G}(u)+\delta _{G}(v)}. \end{aligned}$$

  Thus,

  $$\begin{aligned} 2m^{2} \le GA_{5}(G)\sum \limits _{uv \in E(G)}\frac{\delta _{G}(u)+\delta _{G}(v)}{\sqrt{\delta _{G}(u)\delta _{G}(v)}}. \end{aligned}$$

  (9)

  We know that

  $$\begin{aligned} \frac{\delta _{G}(u)}{\delta _{G}(v)}+\frac{\delta _{G}(v)}{\delta _{G}(u)} \ge \sqrt{\frac{\delta _{G}(u)}{\delta _{G}(v)}}+\sqrt{\frac{\delta _{G}(v)}{\delta _{G}(u)}}. \end{aligned}$$

  From 9, we obtain $2m^{2} \le GA_{5}(G)\mathrm{ND}_{5}(G)$, i.e.

  $$\begin{aligned} \mathrm{ND}_{5}(G) \ge \frac{2m^{2}}{ GA_{5}}. \end{aligned}$$

  Equality holds iff $\frac{\delta _{G}(u)+\delta _{G}(v)}{\sqrt{\delta _{G}(u)\delta _{G}(v)}} = k$, a constant $\forall uv \in E(G)$. That is, $\delta _{G}(u) = {\text {some constant}} \times \delta _{G}(v)$ $\forall uv \in E(G)$, i.e., G is regular or complete bipartite graph.

• (ii) By Cauchy–Schwarz inequality, we have

  $$\begin{aligned} \mathrm{ND}_{5}(G)= & {} \sum \limits _{uv \in E(G)}\frac{[\delta _{G}(u)+\delta _{G}(v)]^{2}}{\delta _{G}(u)\delta _{G}(v)}-2m \\\ge & {} \frac{1}{\Delta _{N}^{2}}\sum \limits _{uv \in E(G)}[\delta _{G}(u)+\delta _{G}(v)]^{2}-2m\\= & {} \frac{1}{m\Delta _{N}^{2}}\sum \limits _{uv \in E(G)}1^{2}\sum \limits _{uv \in E(G)}[\delta _{G}(u)+\delta _{G}(v)]^{2} -2m \\\ge & {} \frac{1}{m\Delta _{N}^{2}}[\sum \limits _{uv \in E(G)}(\delta _{G}(u)+\delta _{G}(v))]^{2}-2m \\= & {} \frac{4M_{2}(G)^{2}}{m\Delta _{N}^{2}}-2m. \end{aligned}$$

Equality holds iff $\delta _{G}(u)=\Delta _{N} = \delta _{G}(v)$ and $\delta _{G}(u)+\delta _{G}(v) = c,$ a constant occur simultaneously for all $uv \in E(G).$ That is, G is regular or complete bipartite graph.

Hence the proof $\square$

It is obvious that, $\delta _G (u)\ge d_G (u)$ and $\delta _G (v) \ge d_G (v)$, $\forall uv \in E(G)$. Equality appears for $P_2$ only. Keeping in mind this fact, we have the following proposition.

Proposition 7

For a graph G, having Randic index R(G), second Zagreb index $M_2 (G)$, reciprocal Randic index $\mathrm{RR}(G)$, sum-connectivity index $\mathrm{SCI}(G)$, we have

• (i) $\mathrm{ND}_{1}(G) \ge \mathrm{RR}(G)$

• (ii) $\mathrm{ND}_{2}(G) \ge \mathrm{SCI}(G)$

• (ii) $\mathrm{ND}_{3}(G) \ge \mathrm{ReZG}_{3}(G)$

• (iii) $\mathrm{ND}_{4}(G) \ge R(G)$

• (iv) $\mathrm{ND}_{5}(G) \le 2M_{2}(G)$

• (v) $\mathrm{ND}_{6}(G) \le 2M_{2}(G)$

Equality holds in each case iff G is $P_2$.

Conclusion

In this article, we have proposed some novel topological indices based on neighborhood degree sum of end vertices of edges. Their predictive ability have tested using octane isomers and alkanes from n-butanes to nonanes. These indices have demonstrated as useful molecular descriptors in QSPR study. These indices are an extension of some well established indices based on degree. The correlations between these new indices and the different properties and activities are often stronger, sometimes slightly weaker than the old indices. For octane isomers, the $\mathrm{ND}_{1}$ index can model acentric factor with high precision compared to the existing indices under consideration. For alkanes, the $\mathrm{ND}_{5}$ index is more effective in predicting ct and st compared to other well-known indices. The predictability of $\mathrm{ND}_2$ index is remarkable for cp compared to the existing and often used topological indices. The sensitivity test (Table 26) confirms the supremacy of the novel indices compared to the old indices. We have also correlated these indices with other degree-based topological indices. This investigation on Tables 24, 25 concludes that $\mathrm{ND}_{5}$ index is independent among all novel indices. This work ends with computing some bounds of these novel indices. For further research, these indices can be computed for various graph operations and some composite graphs and networks.