# On Topological Analysis of Niobium (II) Oxide Network via Curve Fitting and Entropy Measures

Authors: Muhammad Kamran Siddiqui; Sana Javed; Sadia Khalid; Mazhar Hussain; Muhammad Shahbaz; Samuel Asefa Fufa
Journal: Complexity
 (2022)

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## 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.


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## Body

## 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 ofin 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].NiobiumNb, 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, NbO, NbO2, Nb2O5, and other intermediary phases, such as Nb12O29, are useful. Apart from that, niobium oxides have a variety of innovative uses. Niobium monoxide NbO is utilized as a gate electrode in transistors [28], and a NbO/NbO2 junction may be employed in robust switching devices [29]. NbO crystallises in the form of a face-centered cubic structure similar to sodium chloride crystal where every Nb atom in a square planar lattice is linked to four oxygen atoms [30]. Furthermore, the NbO crystal structure is unique in which it has 25 percent arranged voids in both the Nb and O sublattices as shown in Figure 1 [31].Figure 1
Cubic structure ofNbO.Researchers have investigated the electrical and thermophysical properties ofNbO. NbO has a density of around 7.3g/cm3 and a melting temperature of 1940°C [31]. Niobium monoxide exhibits typical metallic behaviour and is usually recognised as a metal, with a resistivity of around 21l×cm at 25°C that drops with temperature to 1.8l×cm at 4.2k. Researchers have measured X-ray fluorescence for several niobium oxides and correlated the findings of NbO to the conduction and valence band calculations of Nb1.0O∞!, discovering substantial variances [32]. They attempted to emulate the NbO structure by doing band structure calculations for Nb0.75O0.75 in order to account for the 25% 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 G=V,E, where La denotes the degree of a vertex a as the number of edges with a:(1)RαG=Rα=∑lm∈EGLl×Lmα,whereα∈1,−1,12,−12,2.Estrada et al. [36, 37] established atom bond connectivity index as follows:(2)ABCG=ABC=∑lm∈EGLl+Lm−2Ll×Lm.Vukičević and Furtula [38] presented the geometric arithmetic index as follows:(3)GAG=GA=∑lm∈EG2Ll×LmLl+Lm.The Zagreb indices defined in [20, 39, 40] are as follows:(4)M1G=M1=∑lm∈EGLl+Lm,M2G=M2=∑lm∈EGLl×Lm.The first and second Zagreb coindices defined in [41, 42] are as follows:(5)M¯1G=M¯1=∑lm∉EGLl+Lm,M¯2G=M¯2=∑lm∉EGLl×Lm.Gutman and Trinajstić [40] and Furtula and Gutman [43] introduced forgotten index as follows:(6)FG=F=∑lm∈EGLl2+Lm2.Wang et al. [44] described the augmented Zagreb index as(7)AZIG=AZI=∑lm∈EGLl×LmLl+Lm−23.The Balaban index [45, 46] is presented as follows:(8)JG=J=ll−m∑lm∈EG1Ll×Lm.Ranjini et al. in [47] reformulated versions of Zagreb indices as follows:(9)ReZG1G=ReZG1=∑lm∈EGLl+LmLl×Lm,ReZG2G=ReZG2=∑lm∈EGLl×LmLl+Lm,ReZG3G=ReZG3=∑lm∈EGLl×LmLl+Lm.


## 2. Results for Niobium (II) Oxide

The number of vertices and edges of structure of Niobium (II) oxide denoted byNbO is 9lm+5l+5m+2 and 16lm+6l+6m, respectively. In NbOthere are three types of vertices, namely, the vertices of degree 2,3, and 4, respectively. The vertex and edge partition of NbO is presented in Table 1 and Table 2, respectively.Table 1
Vertex partition ofNbO.
LvFrequencySet of vertices28V134lm+8l+8m−8V345lm−3l−3m+2V4Table 2
Edge partition ofNbO.
Ll,LmFrequencySet of edges2,316E13,316l+16m−24E23,412lm−8l−8m+8E34,44lm−2l−2mE4Theorem 1.
LetG≅NbOl,m with l,m≥1. Then, Randić indices for α∈1,−1,1/2,−1/2 are as follows:(10)R1=208lm+16l+16m−24,R−1=1.125lm+1.1111l+0.9861m+0.6666,R1/2=49.5692lm+20.2871l+12.2871m−5.0953,R−1/2=3.9641lm+3.0239l+2.5239m+0.8413.Proof.
Forα=1,(11)R1=∑lm∈EGLl×Lm=166+16l+16m−249+12lm−8l−8m+812+4lm−2l−2m16=208lm+16l+16m−24.
Forα=−1,(12)R−1=∑lm∈EG1Ll×Lm=1616+16l+16m−2419+12lm−8l−8m+8112+4lm−2l−2m116=1.125lm+1.1111l+0.9861m+0.6666.
Forα=1/2,(13)R1/2=∑lm∈EGLl×Lm=166+16l+16m−249+12lm−8l−8m+812+4lm−2l−2m16=49.5692lm+20.2871l+12.2871m−5.0953.
Forα=−1/2,(14)R−1/2=∑lm∈EG1Ll×Lm=1616+16l+16m−2419+12lm−8l−8m+8112+4lm−2l−2m116=3.9641lm+3.0239l+2.5239m+0.8413.Theorem 2.
LetG≅NbO, with l,m≥1. Then, the atom bond connectivity index corresponds to(15)ABC=10.1954lm+4.2779l+4.2779m+0.4776.Proof.
(16)ABC=∑lm∈EGLl+Lm−2Ll×Lm=1636+16l+16m−2449+12lm−8l−8m+8512+4lm−2l−2m616=10.1954lm+4.2779l+4.2779m+0.4776.Theorem 3.
Consider the graph ofG≅NbO which has l,m≥1 and geometric arithmetic index is corresponding to the following:(17)GA=15.8769lm+6.0820l+6.0820m−0.4053.Proof.
(18)GA=∑lm∈EG2Ll×LmLl+Lm=16265+16l+16m−24296+12lm−8l−8m+82127+4lm−2l−2m2168=15.8769lm+6.0820l+6.0820m−0.4053.Theorem 4.
The forgotten index for the graph ofG≅NbOl,m with l,m≥1 is corresponding to(19)F=428lm+24l+24m−24.Proof.
(20)F=∑lm∈EGLl2+Lm2=1613+16l+16m−2418+12lm−8l−8m+825+4lm−2l−2m32=428lm+24l+24m−24.Theorem 5.
The augmented index for the graph ofG≅NbOl,m with l,m≥1 is corresponding to(21)AZI=241.7398lm+33.7320l+33.7320m−34.783.Proof.
(22)AZI=∑lm∈EGLl×LmLl+Lm−23=16633+16l+16m−24943+12lm−8l−8m+81253+4lm−2l−2m1663=241.7398lm+33.7320l+33.7320m−34.783.Theorem 6.
Consider the graph ofG≅NbOl,m such that l,m≥1 and the first and second Zagreb index is corresponding to(23)M1=116lm+24l+24m−8,M2=208lm+16l+16m−24.Proof.
(24)M1=∑lm∈EGLl+Lm=165+16l+16m−246+12lm−8l−8m+87+4lm−2l−2m8=116lm+24l+24m−8,M2=∑lm∈EGLl×Lm=166+16l+16m−249+12lm−8l−8m+812+4lm−2l−2m16=208lm+16l+16m−24.Theorem 7.
The first and second Zagreb coindices for the graph ofG≅NbOl,m with l,m≥1 are corresponding to(25)M¯1=288l2m2+268l2m+268lm2+60l2+36lm+60m2−12l−12m+8,M¯2=512l2m2+384l2m+384lm2+72l2−122lm+72m2−28l−28m+28.Proof.
(26)M¯1=∑lm∉EGLl+Lm=2EGVG−1−M1=216lm+6l+6m9lm+5l+5m+2−1−116lm+24l+24m−8=288l2m2+268l2m+268lm2+60l2+36lm+60m2−12l−12m+8,M¯2=∑lm∉EGLl×Lm=2EG2−12M1−M2=216lm+6l+6m2−12116lm+24l+24m−8−208lm+16l+16m−24=512l2m2+384l2m+384lm2+72l2−122lm+72m2−28l−28m+28.Theorem 8.
The redefined Zagreb indices for the graph ofG≅NbIl,m with l,m≥1 correspond to(27)ReZG1=9lm+5l+5m+2,ReZG2=28.5714lm+6.2857l+6.2857m−3.0857,ReZG3=1520lm−64l−64m−144.Proof.
(28)ReZG1=∑lm∈EGLl+LmLl×Lm=1656+16l+16m−2469+12lm−8l−8m+8712+4lm−2l−2m816=9lm+5l+5m+2,ReZG2=∑lm∈EGLl×LmLl+Lm=1665+16l+16m−2496+12lm−8l−8m+8127+4lm−2l−2m168=28.5714lm+6.2857l+6.2857m−3.0857,ReZG3=∑lm∈EGLl×LmLl+Lm=1630+16l+16m−2454+12lm−8l−8m+884+4lm−2l−2m128=1520lm−64l−64m−144.
Graphical illustration for each index corresponding tol=m=i;i=1,2,…,100, computed above, is provided in Figure 2.Figure 2
An interactive view of scattered plot together with surface plot of indices. (a) (m, n,R1). (b) (m,n,R−1). (c) m,n,R1/2. (d) m,n,R−1/2. (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) m,n,M¯1. (l) m,n,M¯2. (m) (m, n, ReZG1). (n) (m, n, ReZG2). (o) (m, n, ReZG3).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)


### 2.1. Thermodynamical Properties (HoF and Entropy) of Niobium (II) Oxide

Many topological indices are derived for unit cell ofNbO, including M1; M1; ABC; GA; and AZI. These indices are linked to thermodynamic properties of NbO, such as heat of formation HoF or enthalpy and entropy. The standard molar enthalpy and entropy of NbO is −405.8kJmol−1 and 48.1Jmol−1K−1, respectively. Table 3 represents the numerical values of HoF and Entropy.Table 3
Values ofHoF and Entropy for units of NbO.
m,nUnits of frequencyHoF×10−22kJEntropy×10−22Jmol−1K−11,14−26.95453.19492,216−107.818012.77983,336−242.590528.75454,464−431.272051.11925,5100−673.862579.87376,6144−970.3620115.01827,7196−1320.7705156.5526


### 2.2. Statistical Models forHoF and Topological Indices

In this section, mathematical frameworks are created for the topological index (computed in Section2) and HoF (given in Section 2.1) of NbO. 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. HoF for NbO is depicted in Table 19. Let ε and γ denote the mean and standard deviation that is used to rescale the data.(i)
Estimation of rational polynomial forHoF vs. R1 is(29)HoFx=p1x+p2x4+q1x3+q2x2+q3x+q4,wherex=R1 is rescaled through ε=4264 and γ=3746.(ii)
Estimation of rational polynomial forHoF vs. R−1 is(30)HoFx=p1x2+p2x+p3x2+q1x+q2,wherex=R−1 is rescaled through ε=31.56 and γ=24.34.(iii)
Estimation of rational polynomial forHoF vs. R1/2 is(31)HoFx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=R1/2 rescaled through ε=1117 and γ=945.4.(iv)
Estimation of rational polynomial forHoF vs. R−1/2 is(32)HoFx=p1x2+p2x+p3x2+q1x+q2,wherex=R−1/2 is rescaled through ε=102.3 and γ=81.85.(v)
Estimation of rational polynomial forHoF vs. ABC is(33)HoFx=p1x+p2x2+q1x+q2,wherex=ABC is rescaled through ε=238.6 and γ=198.4.(vi)
Estimation of rational polynomial forHoF vs. GA is(34)HoFx=p1x2+p2x+p3x+q1,wherex=GA is rescaled through ε=365.8 and γ=306.5.(vii)
Estimation of rational polynomial forHoF vs. M1 is(35)HoFx=p1x+p2x4+q1x3+q2x2+q3x+q4,wherex=M1 is rescaled through ε=2504 and γ=2153.(viii)
Estimation of rational polynomial forHoF vs. M2 is(36)HoFx=p1x+p2x3+q1x2+q2x+q3,wherex=M2 is rescaled through ε=4264 and γ=3746.(ix)
Estimation of rational polynomial forHoF vs. M¯1 is(37)HoFx=p1x2+p2x+p3x2+q1x+q2,wherex=M¯1 is rescaled through ε=2.554e+05 and γ=3.276e+05.(x)
Estimation of rational polynomial forHoF vs. M¯2 is(38)HoFx=p1x2+p2x+p3x2+q1x+q2,wherex=M¯2 is rescaled through ε=4.283e+05 and γ=5.545e+05.(vi)
Estimation of rational polynomial forHoF vs. AZI is(39)HoFx=p1x2+p2x+p3x+q1,wherex=AZI is rescaled through ε=1818 and γ=1625.(vii)
Estimation of rational polynomial forHoF vs. F is(40)HoFx=p1x2+p2x+p3x+q1,wherex=F is rescaled through ε=8728 and γ=7669.(viii)
Estimation of rational polynomial forHoF vs. ReZG1 is(41)HoFx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=ReZG1 is rescaled through ε=222 and γ=180.3.(ix)
Estimation of rational polynomial forHoF vs. ReZG2 is(42)HoFx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=ReZG2 is rescaled through ε=618.6 and γ=531.8.(x)
Estimation of rational polynomial forHoF vs. ReZG3 is(43)HoFx=p1x2+p2x+p3x2+q1x+q2,where x=ReZG3 is rescaled through ε=2.974e+04 and γ=2.661e+04.Figure 3
R1 (x-axis) vs. HoF (y-axis).Figure 4
R−1 vs. HoF.Figure 5
R1/2 vs. HoF.Figure 6
R−1/2 vs. HoF.Figure 7
ABC vs. HoF.Figure 8
GA vs. HoF.Figure 9
M1 vs. HoF.Figure 10
M2 vs. HoF.Figure 11
M¯1 vs. HoF.Figure 12
M¯2 vs. HoF.Figure 13
AZI vs. HoF.Figure 14
F vs. HoF.Figure 15
ReZG1 vs. HoF.Figure 16
ReZG2 vs. HoF.Figure 17
ReZG3 vs. HoF.Table 4
R1 vs. HoF.
piConfidence intervalCIqiCIi=1−2.089e + 05(−1.39e + 06, 9.72e + 05)−3.028(−8.911, 2.856)i=2−2.381e + 05(−1.583e + 06, 1.107e + 06)3.559(−15.22, 22.34)i=3——−4.613(−33.85, 24.62)i=4——443.2(−2061, 2947)Table 5
R−1 vs. HoF.
piCIqiCIi=1−3.496e + 07(−1.177e + 11, 1.176e + 11)6.704e + 04(−2.255e + 08, 2.257e + 08)i=2−9.916e + 07(−3.337e + 11, 3.335e + 11)1.33e + 05(−4.473e + 08, 4.476e + 08)i=3−6.91e + 07(−2.325e + 11, 2.324e + 11)——Table 6
R1/2 vs. HoF.
piCIqiCIi=13.108e + 07(−6.357e + 10, 6.364e + 10)229.4(−4.749e + 05, 4.753e + 05)i=28.362e + 07(−1.71e + 11, 1.711e + 11)−6.361e + 04(−1.302e + 08, 1.301e + 08)i=35.528e + 07(−1.13e + 11, 1.131e + 11)−1.04e + 05(−2.128e + 08, 2.126e + 08)Table 7
R−1/2 vs. HoF.
piCIqiCIi=1−3.55e + 07(−9.978e + 10, 9.971e + 10)6.963e + 04(−1.956e + 08, 1.957e + 08)i=2−9.948e + 07(−2.796e + 11, 2.794e + 11)1.305e + 05(−3.666e + 08, 3.668e + 08)i=3−6.838e + 07(−1.921e + 11, 1.92e + 11)——Table 8
ABC vs. HoF.
piCIqiCIi=1−3.504e + 04(−6.796e + 04, −2111)−3.75(−5.523, −1.977)i=2−4.081e + 04(−7.88e + 04, −2822)77.34(5.61, 149.1)Table 9
GA vs. HoF.
piCIqiCIi=14.637e + 05(−2.16e + 10, 2.16e + 10)−4.433e + 04(−2.063e + 09, 2.063e + 09)i=22.091e + 07(−9.735e + 11, 9.736e + 11)——i=32.35e + 07(−1.094e + 12, 1.094e + 12)——Table 10
M1 vs. HoF.
piCIqiCIi=1−8.239e + 04(−5.395e + 05, 3.747e + 05)−3.054(−8.908, 2.8)i=2−9.506e + 04(−6.215e + 05, 4.314e + 05)3.673(−15.16, 22.51)i=3——−4.831(−34.56, 24.9)i=4——177.9(−807.8, 1164)Table 11
M2 vs. HoF.
piCIqiCIi=1−7.613e + 07(−9.059e + 10, 9.044e + 10)442.4(−5.281e + 05, 5.29e + 05)i=2−8.664e + 07(−1.031e + 11, 1.029e + 11)−1634(−1.945e + 06, 1.942e + 06)i=3——1.614e + 05(−1.917e + 08, 1.921e + 08)Table 12
M¯1 vs. HoF.
piCIqiCIi=1−3053(−4237, −1869)6.184(2.407, 9.962)i=2−6405(−1.054e+04, −2273)4.651(0.886, 8.417)i=3−3146(−5686, −606.9)——Table 13
M¯2 vs. HoF.
piCIqiCIi=11.765e + 07(−2.879e + 12, 2.879e + 12)−7.947e + 04(−1.296e + 10, 1.296e + 10)i=21.026e + 08(−1.673e + 13, 1.673e + 13)−1.008e + 05(−1.644e + 10, 1.644e + 10)i=37.063e + 07(−1.152e + 13, 1.152e + 13)——Table 14
AZI vs. HoF.
piCIqiCIi=1−8568(−1.056e + 08, 1.056e + 08)−1.092e + 04(−1.426e + 08, 1.425e + 08)i=25.208e + 06(−6.799e + 10, 6.8e + 10)——I=35.894e + 06(−7.693e + 10, 7.694e + 10)——Table 15
F vs. HoF.
piCIqiCIi=12.344e + 04(−4.068e + 08, 4.069e + 08)−1.46e + 04(−2.484e + 08, 2.484e + 08)i=26.948e + 06(−1.182e + 11, 1.182e + 11)——i=37.852e + 06(−1.336e + 11, 1.336e + 11)——Table 16
ReZG1 vs. HoF.
piCIqiCIi=14.092e + 07(−4.611e + 11, 4.611e + 11)1129(−1.274e + 07, 1.275e + 07)i=21.012e + 08(−1.14e + 12, 1.14e + 12)−8.418e + 04(−9.487e + 08, 9.485e + 08)i=36.262e + 07(−7.054e + 11, 7.056e + 11)−1.188e + 05(−1.339e + 09, 1.338e + 09)Table 17
ReZG2 vs. HoF.
piCIqiCIi=1−4.999e + 05(−1.692e + 06, 6.922e + 05)−5.973(−14.18, 2.234)i=2−1.264e + 06(−4.372e + 06, 1.845e + 06)1037(−1429, 3503)i=3−7.929e + 05(−2.795e + 06, 1.209e + 06)1486(−2265, 5236)Table 18
ReZG3 vs. HoF.
piCIqiCIi=1−1.746e + 05(−4.898e + 05, 1.407e + 05)364.5(−298.9, 1028)i=2−1.921e + 05(−9.603e + 05, 5.761e + 05)−10.06(−976, 955.8)i=35470(−5.163e + 05, 5.272e + 05)——Table 19
Goodness of fit for nidices vs.HoF for NbO.
IndexFit typeSSER2AdjustedR2RMSER1rat140.02675110.1636R−1rat224.062111.425R1/2rat230.0617110.2484R−1/2rat223.122111.249ABCrat1215.19112.25GArat2161.1110.99994.513M1rat140.1551110.3938M2rat130.7451110.6104M¯1rat2292.310.99990.99986.794M¯2rat2217690.99870.996129.74HMrat130.123110.248AZIrat210.3795110.3557Frat211.776110.7695ReZG1rat232.238111.496ReZG2rat230.0005708110.02389ReZG3rat220.8455110.6502


### 2.3. Statistical Models for Entropy and Topological Indices

In this section, mathematical frameworks for the topological index (computed in Section2) and Entropy (given in Section 2.1) of NbO 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 NbO is depicted in Table 35.Figure 18
R1 (x-axis) vs. Entropy (y-axis).Figure 19
R−1 vs. Entropy.Figure 20
R1/2 vs. Entropy.Figure 21
R−1/2 vs. Entropy.Figure 22
ABC vs. Entropy.Figure 23
GA vs. Entropy.Figure 24
M1 vs. Entropy.Figure 25
M2 vs. Entropy.Figure 26
M¯1 vs. Entropy.Figure 27
M¯2 vs. Entropy.Figure 28
AZI vs. Entropy.Figure 29
F vs. Entropy.Figure 30
ReZG1 vs. Entropy.Figure 31
ReZG2 vs. Entropy.Figure 32
ReZG3 vs. Entropy.Table 20
R1 vs. Entropy.
piCIqiCIi=1−2.646e + 04(−6.984e + 04, 1.691e + 04)−3.384(−5.642, −1.127)i=2−3.014e + 04(−7.949e + 04, 1.921e + 04)5.483(−2.395, 13.36)i=3——−473.4(−1249, 302.2)Table 21
R−1 vs. Entropy.
piCIqiCIi=1−7.6e + 05(−4.028e + 08, 4.013e + 08)135.2(−7.287e + 04, 7.314e + 04)i=2−2.077e + 06(−1.1e + 09, 1.096e + 09)−1.259e + 04(−6.675e + 06, 6.65e + 06)i=3−1.406e + 06(−7.444e + 08, 7.415e + 08)−2.283e + 04(−1.208e + 07, 1.204e + 07)Table 22
R1/2 vs. Entropy.
piCIqiCIi=1−1.575e + 06(−1.165e + 09, 1.162e + 09)102.9(−7.795e + 04, 7.816e + 04)i=2−4.215e + 06(−3.115e + 09, 3.106e + 09)−2.721e + 04(−2.013e + 07, 2.008e + 07)i=3−2.775e + 06(−2.049e + 09, 2.044e + 09)−4.406e + 04(−3.254e + 07, 3.245e + 07)Table 23
R−1/2 vs. Entropy.
piCIqiCIi=12.454e + 04(−2.946e + 04, 7.854e + 04)−6.267(−14.22, 1.686)i=26.443e + 04(−8.139e + 04, 2.102e + 05)416.8(−493.2, 1327)i=34.205e + 04(−5.534e + 04, 1.394e + 05)677.3(−891.1, 2246)Table 24
ABC vs. Entropy.
piCIqiCIi=18.579e + 05(−2.287e + 10, 2.287e + 10)308(−8.305e + 06, 8.306e + 06)i=22.692e + 06(−7.184e + 10, 7.185e + 10)1.379e + 04(−3.674e + 08, 3.675e + 08)i=31.993e + 06(−5.323e + 10, 5.323e + 10)3.185e + 04(−8.508e + 08, 8.508e + 08)Table 25
GA vs. Entropy.
piCIqiCIi=1−5810(−1.493e + 04, 3315)−3.43(−5.625, −1.235)i=2−6784(−1.739e + 04, 3818)5.695(−2.055, 13.45)i=3——−108(−277.2, 61.22)Table 26
M1 vs. Entropy.
piCIqiCIi=19765(−4.435e + 04, 6.388e + 04)−3.054(−8.9, 2.792)i=21.127e + 04(−5.105e + 04, 7.358e + 04)3.672(−15.14, 22.48)i=3——−4.83(−34.52, 24.86)i=4——177.9(−806.6, 1162)Table 27
M2 vs. Entropy.
piCIqiCIi=12.117e + 04(421.2, 4.192e + 04)−3.726(−5.577, −1.875)i=22.408e + 04(522.6, 4.765e + 04)378.5(8.47, 748.6)Table 28
M¯1 vs. Entropy.
piCIqiCIi=1−6.943e + 06(−2.572e + 11, 2.572e + 11)4530(−1.679e + 08, 1.679e + 08)i=2−5.788e + 06(−2.144e + 11, 2.144e + 11)−3.523e + 04(−1.305e + 09, 1.305e + 09)i=3——−7.099e + 04(−2.63e + 09, 2.63e + 09)Table 29
M¯2 vs. Entropy.
piCIqiCIi=14.171e + 06(−3.796e + 11, 3.796e + 11)3482(−3.171e + 08, 3.171e + 08)i=23.268e + 06(−2.975e + 11, 2.975e + 11)−1.149e + 04(−1.045e + 09, 1.045e + 09)i=3——2.548e + 04(−2.319e + 09, 2.319e + 09)i=4——4.081e + 04(−3.714e + 09, 3.715e + 09)Table 30
AZI vs. Entropy.
piCIqiCIi=15.214e + 06(−1.034e + 09, 1.044e + 09)166.1(−3.334e + 04, 3.367e + 04)i=25.886e + 06(−1.167e + 09, 1.179e + 09)9.2e + 04(−1.825e + 07, 1.843e + 07)Table 31
F vs. Entropy.
piCIqiCIi=13.363e + 04(−1.563e + 05, 2.236e + 05)−3.023(−8.886, 2.84)i=23.825e + 04(−1.777e + 05, 2.542e + 05)3.538(−15.15, 22.22)i=3——−4.576(−33.6, 24.45)i=4——600.1(−2788, 3988)Table 32
ReZG1 vs. Entropy.
piCIqiCIi=12.858e + 04(−3.576e + 04, 9.292e + 04)−6.187(−14.22, 1.843)i=27.425e + 04(−9.781e + 04, 2.463e + 05)490.1(−606, 1586)i=34.793e + 04(−6.582e + 04, 1.617e + 05)768.1(−1054, 2590)Table 33
ReZG2 vs. Entropy.
piCIqiCIi=15.915e + 04(−7.923e + 04, 1.975e + 05)−5.968(−14.01, 2.073)i=21.495e + 05(−2.114e + 05, 5.104e + 05)1035(−1380, 3450)i=39.381e + 04(−1.386e + 05, 3.262e + 05)1483(−2190, 5155)Table 34
ReZG3 vs. Entropy.
piCIqiCIi=18.157e + 04(3.2e + 04, 1.311e + 05)1448(566.3, 2329)i=22.083e + 05(7.129e + 04, 3.454e + 05)2048(618.7, 3478)i=31.311e + 05(3.961e + 04, 2.227e + 05)——Table 35
Goodness of fit for indices vs. entropy forNb0.
IndexFit typeSSER2AdjustedR2RMSER1rat130.002287110.03382R−1rat230.001106110.03325R1/2rat230.0006468110.02543R−1/2rat232.69e−05110.005187ABCrat230.722510.99980.85GArat130.0376110.1371M1rat140.002174110.04663M2rat120.01024110.05843M¯1rat1350.830.99730.9925.041M¯2rat146.750.99960.99792.598HMrat239.678e−07110.0009838AZIrat120.007279110.04926Frat140.0002058110.01434ReZG1rat232.314e−05110.00481ReZG2rat237.752e−06110.002784ReZG3rat221.334e−05110.002582(i)
Estimated rational polynomial ofEntropy vs. R1 is(44)Entropyx=p1x+p2x3+q1x2+q2x+q3,wherex=R1 is rescaled through ε=4264 and γ=3746.(ii)
Estimated rational polynomial ofEntropy vs. R−1 is(45)Entropyx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=R−1 is rescaled through ε=31.56 and γ=24.34.(iii)
Estimated rational polynomial ofEntropy vs. R1/2 is(46)Entropyx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=R1/2 is rescaled through ε=1117 and γ=945.4.(iv)
Estimated rational polynomial ofEntropy vs. R−1/2 is(47)Entropyx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=R−1/2 is rescaled through ε=102.3 and γ=81.85.(v)
Estimated rational polynomial ofEntropy vs. ABC is(48)Entropyx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=ABC is rescaled through ε=238.6 and γ=198.4.(vi)
Estimated rational polynomial ofEntropy vs. GA is(49)Entropyx=p1x+p2x3+q1x2+q2x+q3,wherex=GA is rescaled through ε=365.8 and γ=306.5.(vii)
Estimated rational polynomial ofEntropy vs. M1 is(50)Entropyx=p1x+p2x4+q1x3+q2x2+q3x+q4,wherex=M1 is rescaled through ε=2504 and γ=2153.(viii)
Estimated rational polynomial ofEntropy vs. M2 is(51)Entropyx=p1x+p2x2+q1x+q2,wherex=M2 is rescaled through ε=4264 and γ=3746.(ix)
Estimated rational polynomial ofEntropy vs. M¯1 is(52)Entropyx=p1x+p2x3+q1x2+q2x+q3,wherex=M¯1 is rescaled through ε=2.554e+05 and γ=3.276e+05.(x)
Estimated rational polynomial ofEntropy vs. M¯2 is(53)Entropyx=p1x+p2x4+q1x3+q2x2+q3x+q4,wherex=M¯2 is rescaled through ε=4.283e+05 and γ=5.545e+05.(xi)
Estimated rational polynomial ofEntropy vs. AZI is(54)Entropyx=p1x+p2x2+q1x+q2,wherex=AZI is rescaled through ε=1818 and γ=1625.(xii)
Estimated rational polynomial ofEntropy vs. F is(55)Entropyx=p1x+p2x4+q1x3+q2x2+q3x+q4,wherex=F is rescaled through ε=8728 and γ=7669.(xiii)
Estimated rational polynomial ofEntropy vs. ReZG1 is(56)Entropyx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=ReZG1 is rescaled through ε=222 and γ=180.3.(xiv)
Estimated rational polynomial ofEntropy vs. ReZG2 is(57)Entropyx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=ReZG2 is rescaled through ε=618.6 and γ=531.8.(xv)
Estimated rational polynomial ofEntropy vs. ReZG3 is(58)Entropyx=p1x2+p2x+p3x2+q1x+q2,where x=ReZG3 is rescaled through ε=2.974e+04 and γ=2.661e+04.


## 2.1. Thermodynamical Properties (HoF and Entropy) of Niobium (II) Oxide

Many topological indices are derived for unit cell ofNbO, including M1; M1; ABC; GA; and AZI. These indices are linked to thermodynamic properties of NbO, such as heat of formation HoF or enthalpy and entropy. The standard molar enthalpy and entropy of NbO is −405.8kJmol−1 and 48.1Jmol−1K−1, respectively. Table 3 represents the numerical values of HoF and Entropy.Table 3
Values ofHoF and Entropy for units of NbO.
m,nUnits of frequencyHoF×10−22kJEntropy×10−22Jmol−1K−11,14−26.95453.19492,216−107.818012.77983,336−242.590528.75454,464−431.272051.11925,5100−673.862579.87376,6144−970.3620115.01827,7196−1320.7705156.5526


## 2.2. Statistical Models forHoF and Topological Indices

In this section, mathematical frameworks are created for the topological index (computed in Section2) and HoF (given in Section 2.1) of NbO. 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. HoF for NbO is depicted in Table 19. Let ε and γ denote the mean and standard deviation that is used to rescale the data.(i)
Estimation of rational polynomial forHoF vs. R1 is(29)HoFx=p1x+p2x4+q1x3+q2x2+q3x+q4,wherex=R1 is rescaled through ε=4264 and γ=3746.(ii)
Estimation of rational polynomial forHoF vs. R−1 is(30)HoFx=p1x2+p2x+p3x2+q1x+q2,wherex=R−1 is rescaled through ε=31.56 and γ=24.34.(iii)
Estimation of rational polynomial forHoF vs. R1/2 is(31)HoFx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=R1/2 rescaled through ε=1117 and γ=945.4.(iv)
Estimation of rational polynomial forHoF vs. R−1/2 is(32)HoFx=p1x2+p2x+p3x2+q1x+q2,wherex=R−1/2 is rescaled through ε=102.3 and γ=81.85.(v)
Estimation of rational polynomial forHoF vs. ABC is(33)HoFx=p1x+p2x2+q1x+q2,wherex=ABC is rescaled through ε=238.6 and γ=198.4.(vi)
Estimation of rational polynomial forHoF vs. GA is(34)HoFx=p1x2+p2x+p3x+q1,wherex=GA is rescaled through ε=365.8 and γ=306.5.(vii)
Estimation of rational polynomial forHoF vs. M1 is(35)HoFx=p1x+p2x4+q1x3+q2x2+q3x+q4,wherex=M1 is rescaled through ε=2504 and γ=2153.(viii)
Estimation of rational polynomial forHoF vs. M2 is(36)HoFx=p1x+p2x3+q1x2+q2x+q3,wherex=M2 is rescaled through ε=4264 and γ=3746.(ix)
Estimation of rational polynomial forHoF vs. M¯1 is(37)HoFx=p1x2+p2x+p3x2+q1x+q2,wherex=M¯1 is rescaled through ε=2.554e+05 and γ=3.276e+05.(x)
Estimation of rational polynomial forHoF vs. M¯2 is(38)HoFx=p1x2+p2x+p3x2+q1x+q2,wherex=M¯2 is rescaled through ε=4.283e+05 and γ=5.545e+05.(vi)
Estimation of rational polynomial forHoF vs. AZI is(39)HoFx=p1x2+p2x+p3x+q1,wherex=AZI is rescaled through ε=1818 and γ=1625.(vii)
Estimation of rational polynomial forHoF vs. F is(40)HoFx=p1x2+p2x+p3x+q1,wherex=F is rescaled through ε=8728 and γ=7669.(viii)
Estimation of rational polynomial forHoF vs. ReZG1 is(41)HoFx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=ReZG1 is rescaled through ε=222 and γ=180.3.(ix)
Estimation of rational polynomial forHoF vs. ReZG2 is(42)HoFx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=ReZG2 is rescaled through ε=618.6 and γ=531.8.(x)
Estimation of rational polynomial forHoF vs. ReZG3 is(43)HoFx=p1x2+p2x+p3x2+q1x+q2,where x=ReZG3 is rescaled through ε=2.974e+04 and γ=2.661e+04.Figure 3
R1 (x-axis) vs. HoF (y-axis).Figure 4
R−1 vs. HoF.Figure 5
R1/2 vs. HoF.Figure 6
R−1/2 vs. HoF.Figure 7
ABC vs. HoF.Figure 8
GA vs. HoF.Figure 9
M1 vs. HoF.Figure 10
M2 vs. HoF.Figure 11
M¯1 vs. HoF.Figure 12
M¯2 vs. HoF.Figure 13
AZI vs. HoF.Figure 14
F vs. HoF.Figure 15
ReZG1 vs. HoF.Figure 16
ReZG2 vs. HoF.Figure 17
ReZG3 vs. HoF.Table 4
R1 vs. HoF.
piConfidence intervalCIqiCIi=1−2.089e + 05(−1.39e + 06, 9.72e + 05)−3.028(−8.911, 2.856)i=2−2.381e + 05(−1.583e + 06, 1.107e + 06)3.559(−15.22, 22.34)i=3——−4.613(−33.85, 24.62)i=4——443.2(−2061, 2947)Table 5
R−1 vs. HoF.
piCIqiCIi=1−3.496e + 07(−1.177e + 11, 1.176e + 11)6.704e + 04(−2.255e + 08, 2.257e + 08)i=2−9.916e + 07(−3.337e + 11, 3.335e + 11)1.33e + 05(−4.473e + 08, 4.476e + 08)i=3−6.91e + 07(−2.325e + 11, 2.324e + 11)——Table 6
R1/2 vs. HoF.
piCIqiCIi=13.108e + 07(−6.357e + 10, 6.364e + 10)229.4(−4.749e + 05, 4.753e + 05)i=28.362e + 07(−1.71e + 11, 1.711e + 11)−6.361e + 04(−1.302e + 08, 1.301e + 08)i=35.528e + 07(−1.13e + 11, 1.131e + 11)−1.04e + 05(−2.128e + 08, 2.126e + 08)Table 7
R−1/2 vs. HoF.
piCIqiCIi=1−3.55e + 07(−9.978e + 10, 9.971e + 10)6.963e + 04(−1.956e + 08, 1.957e + 08)i=2−9.948e + 07(−2.796e + 11, 2.794e + 11)1.305e + 05(−3.666e + 08, 3.668e + 08)i=3−6.838e + 07(−1.921e + 11, 1.92e + 11)——Table 8
ABC vs. HoF.
piCIqiCIi=1−3.504e + 04(−6.796e + 04, −2111)−3.75(−5.523, −1.977)i=2−4.081e + 04(−7.88e + 04, −2822)77.34(5.61, 149.1)Table 9
GA vs. HoF.
piCIqiCIi=14.637e + 05(−2.16e + 10, 2.16e + 10)−4.433e + 04(−2.063e + 09, 2.063e + 09)i=22.091e + 07(−9.735e + 11, 9.736e + 11)——i=32.35e + 07(−1.094e + 12, 1.094e + 12)——Table 10
M1 vs. HoF.
piCIqiCIi=1−8.239e + 04(−5.395e + 05, 3.747e + 05)−3.054(−8.908, 2.8)i=2−9.506e + 04(−6.215e + 05, 4.314e + 05)3.673(−15.16, 22.51)i=3——−4.831(−34.56, 24.9)i=4——177.9(−807.8, 1164)Table 11
M2 vs. HoF.
piCIqiCIi=1−7.613e + 07(−9.059e + 10, 9.044e + 10)442.4(−5.281e + 05, 5.29e + 05)i=2−8.664e + 07(−1.031e + 11, 1.029e + 11)−1634(−1.945e + 06, 1.942e + 06)i=3——1.614e + 05(−1.917e + 08, 1.921e + 08)Table 12
M¯1 vs. HoF.
piCIqiCIi=1−3053(−4237, −1869)6.184(2.407, 9.962)i=2−6405(−1.054e+04, −2273)4.651(0.886, 8.417)i=3−3146(−5686, −606.9)——Table 13
M¯2 vs. HoF.
piCIqiCIi=11.765e + 07(−2.879e + 12, 2.879e + 12)−7.947e + 04(−1.296e + 10, 1.296e + 10)i=21.026e + 08(−1.673e + 13, 1.673e + 13)−1.008e + 05(−1.644e + 10, 1.644e + 10)i=37.063e + 07(−1.152e + 13, 1.152e + 13)——Table 14
AZI vs. HoF.
piCIqiCIi=1−8568(−1.056e + 08, 1.056e + 08)−1.092e + 04(−1.426e + 08, 1.425e + 08)i=25.208e + 06(−6.799e + 10, 6.8e + 10)——I=35.894e + 06(−7.693e + 10, 7.694e + 10)——Table 15
F vs. HoF.
piCIqiCIi=12.344e + 04(−4.068e + 08, 4.069e + 08)−1.46e + 04(−2.484e + 08, 2.484e + 08)i=26.948e + 06(−1.182e + 11, 1.182e + 11)——i=37.852e + 06(−1.336e + 11, 1.336e + 11)——Table 16
ReZG1 vs. HoF.
piCIqiCIi=14.092e + 07(−4.611e + 11, 4.611e + 11)1129(−1.274e + 07, 1.275e + 07)i=21.012e + 08(−1.14e + 12, 1.14e + 12)−8.418e + 04(−9.487e + 08, 9.485e + 08)i=36.262e + 07(−7.054e + 11, 7.056e + 11)−1.188e + 05(−1.339e + 09, 1.338e + 09)Table 17
ReZG2 vs. HoF.
piCIqiCIi=1−4.999e + 05(−1.692e + 06, 6.922e + 05)−5.973(−14.18, 2.234)i=2−1.264e + 06(−4.372e + 06, 1.845e + 06)1037(−1429, 3503)i=3−7.929e + 05(−2.795e + 06, 1.209e + 06)1486(−2265, 5236)Table 18
ReZG3 vs. HoF.
piCIqiCIi=1−1.746e + 05(−4.898e + 05, 1.407e + 05)364.5(−298.9, 1028)i=2−1.921e + 05(−9.603e + 05, 5.761e + 05)−10.06(−976, 955.8)i=35470(−5.163e + 05, 5.272e + 05)——Table 19
Goodness of fit for nidices vs.HoF for NbO.
IndexFit typeSSER2AdjustedR2RMSER1rat140.02675110.1636R−1rat224.062111.425R1/2rat230.0617110.2484R−1/2rat223.122111.249ABCrat1215.19112.25GArat2161.1110.99994.513M1rat140.1551110.3938M2rat130.7451110.6104M¯1rat2292.310.99990.99986.794M¯2rat2217690.99870.996129.74HMrat130.123110.248AZIrat210.3795110.3557Frat211.776110.7695ReZG1rat232.238111.496ReZG2rat230.0005708110.02389ReZG3rat220.8455110.6502


## 2.3. Statistical Models for Entropy and Topological Indices

In this section, mathematical frameworks for the topological index (computed in Section2) and Entropy (given in Section 2.1) of NbO 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 NbO is depicted in Table 35.Figure 18
R1 (x-axis) vs. Entropy (y-axis).Figure 19
R−1 vs. Entropy.Figure 20
R1/2 vs. Entropy.Figure 21
R−1/2 vs. Entropy.Figure 22
ABC vs. Entropy.Figure 23
GA vs. Entropy.Figure 24
M1 vs. Entropy.Figure 25
M2 vs. Entropy.Figure 26
M¯1 vs. Entropy.Figure 27
M¯2 vs. Entropy.Figure 28
AZI vs. Entropy.Figure 29
F vs. Entropy.Figure 30
ReZG1 vs. Entropy.Figure 31
ReZG2 vs. Entropy.Figure 32
ReZG3 vs. Entropy.Table 20
R1 vs. Entropy.
piCIqiCIi=1−2.646e + 04(−6.984e + 04, 1.691e + 04)−3.384(−5.642, −1.127)i=2−3.014e + 04(−7.949e + 04, 1.921e + 04)5.483(−2.395, 13.36)i=3——−473.4(−1249, 302.2)Table 21
R−1 vs. Entropy.
piCIqiCIi=1−7.6e + 05(−4.028e + 08, 4.013e + 08)135.2(−7.287e + 04, 7.314e + 04)i=2−2.077e + 06(−1.1e + 09, 1.096e + 09)−1.259e + 04(−6.675e + 06, 6.65e + 06)i=3−1.406e + 06(−7.444e + 08, 7.415e + 08)−2.283e + 04(−1.208e + 07, 1.204e + 07)Table 22
R1/2 vs. Entropy.
piCIqiCIi=1−1.575e + 06(−1.165e + 09, 1.162e + 09)102.9(−7.795e + 04, 7.816e + 04)i=2−4.215e + 06(−3.115e + 09, 3.106e + 09)−2.721e + 04(−2.013e + 07, 2.008e + 07)i=3−2.775e + 06(−2.049e + 09, 2.044e + 09)−4.406e + 04(−3.254e + 07, 3.245e + 07)Table 23
R−1/2 vs. Entropy.
piCIqiCIi=12.454e + 04(−2.946e + 04, 7.854e + 04)−6.267(−14.22, 1.686)i=26.443e + 04(−8.139e + 04, 2.102e + 05)416.8(−493.2, 1327)i=34.205e + 04(−5.534e + 04, 1.394e + 05)677.3(−891.1, 2246)Table 24
ABC vs. Entropy.
piCIqiCIi=18.579e + 05(−2.287e + 10, 2.287e + 10)308(−8.305e + 06, 8.306e + 06)i=22.692e + 06(−7.184e + 10, 7.185e + 10)1.379e + 04(−3.674e + 08, 3.675e + 08)i=31.993e + 06(−5.323e + 10, 5.323e + 10)3.185e + 04(−8.508e + 08, 8.508e + 08)Table 25
GA vs. Entropy.
piCIqiCIi=1−5810(−1.493e + 04, 3315)−3.43(−5.625, −1.235)i=2−6784(−1.739e + 04, 3818)5.695(−2.055, 13.45)i=3——−108(−277.2, 61.22)Table 26
M1 vs. Entropy.
piCIqiCIi=19765(−4.435e + 04, 6.388e + 04)−3.054(−8.9, 2.792)i=21.127e + 04(−5.105e + 04, 7.358e + 04)3.672(−15.14, 22.48)i=3——−4.83(−34.52, 24.86)i=4——177.9(−806.6, 1162)Table 27
M2 vs. Entropy.
piCIqiCIi=12.117e + 04(421.2, 4.192e + 04)−3.726(−5.577, −1.875)i=22.408e + 04(522.6, 4.765e + 04)378.5(8.47, 748.6)Table 28
M¯1 vs. Entropy.
piCIqiCIi=1−6.943e + 06(−2.572e + 11, 2.572e + 11)4530(−1.679e + 08, 1.679e + 08)i=2−5.788e + 06(−2.144e + 11, 2.144e + 11)−3.523e + 04(−1.305e + 09, 1.305e + 09)i=3——−7.099e + 04(−2.63e + 09, 2.63e + 09)Table 29
M¯2 vs. Entropy.
piCIqiCIi=14.171e + 06(−3.796e + 11, 3.796e + 11)3482(−3.171e + 08, 3.171e + 08)i=23.268e + 06(−2.975e + 11, 2.975e + 11)−1.149e + 04(−1.045e + 09, 1.045e + 09)i=3——2.548e + 04(−2.319e + 09, 2.319e + 09)i=4——4.081e + 04(−3.714e + 09, 3.715e + 09)Table 30
AZI vs. Entropy.
piCIqiCIi=15.214e + 06(−1.034e + 09, 1.044e + 09)166.1(−3.334e + 04, 3.367e + 04)i=25.886e + 06(−1.167e + 09, 1.179e + 09)9.2e + 04(−1.825e + 07, 1.843e + 07)Table 31
F vs. Entropy.
piCIqiCIi=13.363e + 04(−1.563e + 05, 2.236e + 05)−3.023(−8.886, 2.84)i=23.825e + 04(−1.777e + 05, 2.542e + 05)3.538(−15.15, 22.22)i=3——−4.576(−33.6, 24.45)i=4——600.1(−2788, 3988)Table 32
ReZG1 vs. Entropy.
piCIqiCIi=12.858e + 04(−3.576e + 04, 9.292e + 04)−6.187(−14.22, 1.843)i=27.425e + 04(−9.781e + 04, 2.463e + 05)490.1(−606, 1586)i=34.793e + 04(−6.582e + 04, 1.617e + 05)768.1(−1054, 2590)Table 33
ReZG2 vs. Entropy.
piCIqiCIi=15.915e + 04(−7.923e + 04, 1.975e + 05)−5.968(−14.01, 2.073)i=21.495e + 05(−2.114e + 05, 5.104e + 05)1035(−1380, 3450)i=39.381e + 04(−1.386e + 05, 3.262e + 05)1483(−2190, 5155)Table 34
ReZG3 vs. Entropy.
piCIqiCIi=18.157e + 04(3.2e + 04, 1.311e + 05)1448(566.3, 2329)i=22.083e + 05(7.129e + 04, 3.454e + 05)2048(618.7, 3478)i=31.311e + 05(3.961e + 04, 2.227e + 05)——Table 35
Goodness of fit for indices vs. entropy forNb0.
IndexFit typeSSER2AdjustedR2RMSER1rat130.002287110.03382R−1rat230.001106110.03325R1/2rat230.0006468110.02543R−1/2rat232.69e−05110.005187ABCrat230.722510.99980.85GArat130.0376110.1371M1rat140.002174110.04663M2rat120.01024110.05843M¯1rat1350.830.99730.9925.041M¯2rat146.750.99960.99792.598HMrat239.678e−07110.0009838AZIrat120.007279110.04926Frat140.0002058110.01434ReZG1rat232.314e−05110.00481ReZG2rat237.752e−06110.002784ReZG3rat221.334e−05110.002582(i)
Estimated rational polynomial ofEntropy vs. R1 is(44)Entropyx=p1x+p2x3+q1x2+q2x+q3,wherex=R1 is rescaled through ε=4264 and γ=3746.(ii)
Estimated rational polynomial ofEntropy vs. R−1 is(45)Entropyx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=R−1 is rescaled through ε=31.56 and γ=24.34.(iii)
Estimated rational polynomial ofEntropy vs. R1/2 is(46)Entropyx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=R1/2 is rescaled through ε=1117 and γ=945.4.(iv)
Estimated rational polynomial ofEntropy vs. R−1/2 is(47)Entropyx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=R−1/2 is rescaled through ε=102.3 and γ=81.85.(v)
Estimated rational polynomial ofEntropy vs. ABC is(48)Entropyx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=ABC is rescaled through ε=238.6 and γ=198.4.(vi)
Estimated rational polynomial ofEntropy vs. GA is(49)Entropyx=p1x+p2x3+q1x2+q2x+q3,wherex=GA is rescaled through ε=365.8 and γ=306.5.(vii)
Estimated rational polynomial ofEntropy vs. M1 is(50)Entropyx=p1x+p2x4+q1x3+q2x2+q3x+q4,wherex=M1 is rescaled through ε=2504 and γ=2153.(viii)
Estimated rational polynomial ofEntropy vs. M2 is(51)Entropyx=p1x+p2x2+q1x+q2,wherex=M2 is rescaled through ε=4264 and γ=3746.(ix)
Estimated rational polynomial ofEntropy vs. M¯1 is(52)Entropyx=p1x+p2x3+q1x2+q2x+q3,wherex=M¯1 is rescaled through ε=2.554e+05 and γ=3.276e+05.(x)
Estimated rational polynomial ofEntropy vs. M¯2 is(53)Entropyx=p1x+p2x4+q1x3+q2x2+q3x+q4,wherex=M¯2 is rescaled through ε=4.283e+05 and γ=5.545e+05.(xi)
Estimated rational polynomial ofEntropy vs. AZI is(54)Entropyx=p1x+p2x2+q1x+q2,wherex=AZI is rescaled through ε=1818 and γ=1625.(xii)
Estimated rational polynomial ofEntropy vs. F is(55)Entropyx=p1x+p2x4+q1x3+q2x2+q3x+q4,wherex=F is rescaled through ε=8728 and γ=7669.(xiii)
Estimated rational polynomial ofEntropy vs. ReZG1 is(56)Entropyx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=ReZG1 is rescaled through ε=222 and γ=180.3.(xiv)
Estimated rational polynomial ofEntropy vs. ReZG2 is(57)Entropyx=p1x2+p2x+p3x3+q1x2+q2x+q3,wherex=ReZG2 is rescaled through ε=618.6 and γ=531.8.(xv)
Estimated rational polynomial ofEntropy vs. ReZG3 is(58)Entropyx=p1x2+p2x+p3x2+q1x+q2,where x=ReZG3 is rescaled through ε=2.974e+04 and γ=2.661e+04.


## 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.



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