Abstract

The remarkable optical features of metallic nanoparticles have extensively developed the interest of scientists and researchers. The generated heat overwhelms cancer tissue incident to nanoparticles with no damage to sound tissues. Niobium nanoparticles have the ability of easy ligands connection so they are very suitable in treating cancer optothermally. A modern field of applied chemistry is chemical graph theory. With the use of combinatorial methods, such as vertex and edge partitions, we explore the connection between atoms and bonds. Topological indices play a vital part in equipping directions to treat cancers or tumors. These indices might be derived experimentally or computed numerically. Although experimental results are worthful but they are expensive as well, so computational analysis provides an economical and rapid way. A topological index is a numerical value that is only determined by the graph. In this paper, we will discuss the chemical graph of niobium (II) oxide. Additionally, each topological index is related with thermodynamical properties of niobium (II) oxide, including entropy and enthalpy. This has been done in MATLAB software, using rational built-in method.

1. Introduction

All types of data quantitative, qualitative, processed, or unprocessed might be considered to gain information to address a simple or a complicated event or situation. If we consider the flow chart of the information, then on the top of the hierarchy we would find notion being the first qualitative obscure assessment of information. The central part of this flow chart comprises of the parameters and measurements while decision making extracted from inference is the final step. Different properties of a chemical compound like its nature, atoms, or chemical state provide us chemical information about the structure [1]. Different chemical reactions in a substance environment produce different physicochemical properties/activities that include boiling point, entropy, heat of formation, or density. In this way, the whole milieu of a substance becomes a promising root of information for the analysis of its chemistry [1]. Supplementary knowledge might be gained by in silico trials for the designing of new compounds for a specified study or objective. Stimulation in such approaches has been seen due to expensive experimental studies along with rigorous biotic and ecological regulations [2]. Such in silico studies are very progressive in medicine due to their cost effectiveness.

Different approaches including graphical quantitative/quantitative structure activity/property relationship (QSAR/QSPR) and modeling have become an essential part of in silico studies in drug development [3, 4]. This is due to the fact that biological variations can be explained in the form of chemical variations. Such analyses are performed continually to obtain profound results [5, 6]. Topological indices play a vital part in equipping directions to treat cancers or tumors. These indices might be derived experimentally or computed numerically [7, 8]. Although experimental results are worthful but they are expensive as well, so computational analysis provides an economical way Recently, several studies are performed/reviewed using the concept of graph theoretical indices in drug research [9, 10]. There is a wide variety of such indices in the literature [11, 12].

A graph usually comprises of two sets, namely, vertex set, that contains the objects, and the edge set; this is based on the connections between the objects. Any chemical compound might be represented in the form of a graph where atoms make the vertex set and the bonding between atoms creates the edge set. Topological indices are based on the atomic connectivity table of the chemical compound [1]. Graphical descriptors, which are usually defined in the form of numerical numbers, can be used to appraise distinct immersed characteristics of a chemical compound from different point of view. Zagreb indices measure the compactness of a molecule so it can be correlated with the physicochemical properties of a compound which depend on the volume/surface ratio of the molecules.

The remarkable optical features of metallic nanoparticles have extensively developed the interest of scientists and researchers [13–15]. Researchers have analyzed that the thermoplasmonic features of nanoparticles might be utilized in treating cancers [16–18]. In optothermal cancer tissue therapy, the descendent laser light provokes the frequency of maximum response amplitude of external plasmon of metallic nanoparticles and consequently the immersed energy of descendent light preserves the heat in nanoparticles [19–21]. The generated heat overwhelms cancer tissue incident to nanoparticles with no damage to sound tissues [22, 23]. Niobium nanoparticles have the ability of easy ligands connection so they are very suitable in treating the cancer optothermally [24–26].

Niobium , a recalcitrant metal, is a suitable construction material for the first shell of nuclear fusion reactors [27]. It does, however, have a high affinity for oxygen and carbon, which are found in pyrotechnics and refrigerants such as liquid. Niobium is renowned to interact very efficiently with oxygen as a component for the first barrier. As a result, reliable thermodynamical data on niobium oxides, , , , and other intermediary phases, such as , are useful. Apart from that, niobium oxides have a variety of innovative uses. Niobium monoxide is utilized as a gate electrode in transistors [28], and a junction may be employed in robust switching devices [29]. crystallises in the form of a face-centered cubic structure similar to sodium chloride crystal where every atom in a square planar lattice is linked to four oxygen atoms [30]. Furthermore, the crystal structure is unique in which it has 25 percent arranged voids in both the and sublattices as shown in Figure 1 [31].


Figure 1 
Cubic structure of .

Researchers have investigated the electrical and thermophysical properties of . has a density of around and a melting temperature of [31]. Niobium monoxide exhibits typical metallic behaviour and is usually recognised as a metal, with a resistivity of around at that drops with temperature to at . Researchers have measured X-ray fluorescence for several niobium oxides and correlated the findings of to the conduction and valence band calculations of , discovering substantial variances [32]. They attempted to emulate the structure by doing band structure calculations for in order to account for the vacancy (see Figure 1). However, the investigation pertaining to the thermodynamic data is very scarce. The laboratory work to study these characteristics is limited due to the analytical limitations. Therefore, computational techniques can be applied to estimate their thermodynamic characteristics. Topological study is useful in this regard [31].

Milan Randić presented the following index, namely, General Randić index [33–35] for a graph , where denotes the degree of a vertex as the number of edges with :

Estrada et al. [36, 37] established atom bond connectivity index as follows:

Vukičević and Furtula [38] presented the geometric arithmetic index as follows:

The Zagreb indices defined in [20, 39, 40] are as follows:

The first and second Zagreb coindices defined in [41, 42] are as follows:

Gutman and Trinajstić [40] and Furtula and Gutman [43] introduced forgotten index as follows:

Wang et al. [44] described the augmented Zagreb index as

The Balaban index [45, 46] is presented as follows:

Ranjini et al. in [47] reformulated versions of Zagreb indices as follows:

2. Results for Niobium (II) Oxide

The number of vertices and edges of structure of Niobium (II) oxide denoted by is and , respectively. In there are three types of vertices, namely, the vertices of degree 2,3, and 4, respectively. The vertex and edge partition of is presented in Table 1 and Table 2, respectively.

Table 1 
Vertex partition of .
Table 2 
Edge partition of .

Theorem 1. Let with . Then, Randić indices for are as follows:

Proof. For ,For ,For ,For ,

Theorem 2. Let , with . Then, the atom bond connectivity index corresponds to

Proof.

Theorem 3. Consider the graph of which has and geometric arithmetic index is corresponding to the following:

Proof.

Theorem 4. The forgotten index for the graph of with is corresponding to

Proof.

Theorem 5. The augmented index for the graph of with is corresponding to

Proof.

Theorem 6. Consider the graph of such that and the first and second Zagreb index is corresponding to

Proof.

Theorem 7. The first and second Zagreb coindices for the graph of with are corresponding to

Proof.

Theorem 8. The redefined Zagreb indices for the graph of with correspond to

Proof. Graphical illustration for each index corresponding to , computed above, is provided in Figure 2.

(a)
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(b)
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(h)
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(j)
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(k)
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(l)
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Figure 2 
An interactive view of scattered plot together with surface plot of indices. (a) (m, n,R1). (b) (m,n,R−1). (c) . (d) . (e) (m, n, ABC). (f) (m, n,GA). (g) (m, n, F). (h) (m, n, AZI). (i) (m, n, M1). (j) (m, n, M2). (k) . (l) . (m) (m, n, ReZG1). (n) (m, n, ReZG2). (o) (m, n, ReZG3).
2.1. Thermodynamical Properties ( and ) of Niobium (II) Oxide

Many topological indices are derived for unit cell of , including ; ; ; ; and . These indices are linked to thermodynamic properties of , such as heat of formation or enthalpy and entropy. The standard molar enthalpy and entropy of is and , respectively. Table 3 represents the numerical values of and .

Table 3 
Values of and for units of .
2.2. Statistical Models for and Topological Indices

In this section, mathematical frameworks are created for the topological index (computed in Section 2) and (given in Section 2.1) of . All fitted curves are shown in Figures 3–17 and also the constant quantity values of the fitted curves are depicted in Tables 4–18. Also, the goodness of fit for indices vs. for is depicted in Table 19. Let and denote the mean and standard deviation that is used to rescale the data.(i)Estimation of rational polynomial for vs. is where is rescaled through and .(ii)Estimation of rational polynomial for vs. is where is rescaled through and .(iii)Estimation of rational polynomial for vs. is where rescaled through and .(iv)Estimation of rational polynomial for vs. is where is rescaled through and .(v)Estimation of rational polynomial for vs. is where is rescaled through and .(vi)Estimation of rational polynomial for vs. is where is rescaled through and .(vii)Estimation of rational polynomial for vs. is where is rescaled through and .(viii)Estimation of rational polynomial for vs. is where is rescaled through and .(ix)Estimation of rational polynomial for vs. is where is rescaled through and .(x)Estimation of rational polynomial for vs. is where is rescaled through and .(vi)Estimation of rational polynomial for vs. is where is rescaled through and .(vii)Estimation of rational polynomial for vs. is where is rescaled through and .(viii)Estimation of rational polynomial for vs. is where is rescaled through and .(ix)Estimation of rational polynomial for vs. is where is rescaled through and .(x)Estimation of rational polynomial for vs. iswhere is rescaled through and .


Figure 3 
(x-axis) vs. (y-axis).

Figure 4 
vs. .

Figure 5 
vs. .

Figure 6 
vs. .

Figure 7 
vs. .

Figure 8 
vs. .

Figure 9 
vs. .

Figure 10 
vs. .

Figure 11 
vs. .

Figure 12 
vs. .

Figure 13 
vs. .

Figure 14 
vs. .

Figure 15 
vs. .

Figure 16 
vs. .

Figure 17 
vs. .
Table 4 
vs. .
Table 5 
vs. .
Table 6 
vs. .
Table 7 
vs. .
Table 8 
vs. .
Table 9 
vs. .
Table 10 
vs. .
Table 11 
vs. .
Table 12 
vs. .
Table 13 
vs. .
Table 14 
vs. .
Table 15 
vs. .
Table 16 
vs. .
Table 17 
vs. .
Table 18 
vs. .
Table 19 
Goodness of fit for nidices vs. for .
2.3. Statistical Models for Entropy and Topological Indices

In this section, mathematical frameworks for the topological index (computed in Section 2) and (given in Section 2.1) of are shown. All fitted curves are shown in Figures 18–32, and the parametric values of the fitted curves are given in Tables 20–34. Also, the goodness of fit for indices vs. entropy for is depicted in Table 35.


Figure 18 
(x-axis) vs. (y-axis).

Figure 19 
vs. .

Figure 20 
vs. .

Figure 21 
vs. .

Figure 22 
vs. .

Figure 23 
vs. .

Figure 24 
vs. .

Figure 25 
vs. .

Figure 26 
vs. .

Figure 27 
vs. .

Figure 28 
vs. .

Figure 29 
vs. .

Figure 30 
vs. .

Figure 31 
vs. .

Figure 32 
vs. .
Table 20 
vs. .
Table 21 
vs. .
Table 22 
vs. .
Table 23 
vs. .
Table 24 
vs. .
Table 25 
vs. .
Table 26 
vs. .
Table 27 
vs. .
Table 28 
vs. .
Table 29 
vs. .
Table 30 
vs. .
Table 31 
vs. .
Table 32 
vs. .
Table 33 
vs. .
Table 34 
vs. .
Table 35 
Goodness of fit for indices vs. entropy for .

(i)Estimated rational polynomial of vs. is where is rescaled through and .(ii)Estimated rational polynomial of vs. is where is rescaled through and .(iii)Estimated rational polynomial of vs. is where is rescaled through and .(iv)Estimated rational polynomial of vs. is where is rescaled through and .(v)Estimated rational polynomial of vs. is where is rescaled through and .(vi)Estimated rational polynomial of vs. is where is rescaled through and .(vii)Estimated rational polynomial of vs. is where is rescaled through and .(viii)Estimated rational polynomial of vs. is where is rescaled through and .(ix)Estimated rational polynomial of vs. is where is rescaled through and .(x)Estimated rational polynomial of vs. is where is rescaled through and .(xi)Estimated rational polynomial of vs. is where is rescaled through and .(xii)Estimated rational polynomial of vs. is where is rescaled through and .(xiii)Estimated rational polynomial of vs. is where is rescaled through and .(xiv)Estimated rational polynomial of vs. is where is rescaled through and .(xv)Estimated rational polynomial of vs. iswhere is rescaled through and .

3. Conclusion

After determining the topological degree-based indices, the thermodynamical parameters of niobium (II) oxide are derived. Fitting curves and building mathematical models are used to create a relationship between each index and each thermodynamical property. In MATLAB software, the rational fitting method is utilized as it gives the least mean squared error of all the built-in methods.

Data Availability

The data used to support the findings of this study are cited at relevant places within the text as references.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

This work was equally contributed by all authors.