Neighborhood M-polynomial of titanium compounds

Sourav Mondal; Muhammad Imran; Nilanjan De; Anita Pal

doi:10.1016/j.arabjc.2021.103244

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Original article

2021

:14;

202108

doi:

10.1016/j.arabjc.2021.103244

Neighborhood M-polynomial of titanium compounds

Sourav Mondal^{^a}, Muhammad Imran^{^b,⁎}, Nilanjan De^{^c}, Anita Pal^{^a}

a Department of Mathematics, National Institute of Technology Durgapur, West Bengal 713209, India

b Department of Mathematical Sciences, United Arab Emirates University, Al Ain 15551, United Arab Emirates

c Department of Basic Sciences and Humanities (Mathematics), Calcutta Institute of Engineering and Management, Kolkata, India

⁎Corresponding author. imrandhab@gmail.com (Muhammad Imran)

Received: 2021-4-6, Accepted: 2021-5-30,

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Abstract

The neighborhood M-polynomial is effective in recovering neighborhood degree sum based topological indices that predict different physical, chemical and biological characteristics of material under investigation. In this work, the neighborhood M-polynomial of Titania nanotube ${TiO}_{2}$ and the crystallographic structure of ${TiF}_{2}$ are obtained. From the neighborhood M-polynomial, some neighborhood degree sum based topological indices are recovered. Effect of oxygen vacancies on outcomes is discussed. A comparative study among the findings and some well-established degree-based indices is performed.

Keywords

Graph

Topological index

Neighborhood M-polynomial

Titania nanotube TiO2

Crystallographic structure TiF2

Primary: 05C35

Secondary: 05C07

05C40

1 Introduction

Throughout this article, we consider connected graph without loops and parallel edges. Let $V (ϒ)$ and $E (ϒ)$ be node and edge sets of a graph $ϒ$ , accordingly. The degree (or valency) of a node $v \in V (ϒ)$ , denoted by $d_{v}$ , is the count of edges incident to v. Here $δ_{u}$ denotes the degree sum of neighbors of u in $ϒ$ . By neighbors of a node, we mean the nodes adjacent to that node. Different properties and activities of a compound are closely related to its molecular graph and this issue is the main topic of interest in chemical graph theory. Quantitative structure–property/ quantitative structure–activity relationships (QSPR/QSAR) are mathematical models which relates the physico-chemical property/ biological activity of compounds to their chemical structures. It has been widely used as an important key field in drug discovery process for predicting physicochemical properties and biological activity of molecules. It expresses the characteristics of molecule through topological index without involving a wet lab. Topological index is a mapping from the collection of graphs to the set of real numbers that describe the topology of graph and are used in QSPR/QSAR analysis. It remains unchanged for isomorphic graphs. Topological indices can be computed by there usual definitions which is laborious while one intends to derive many indices of a certain category. To overcome this approach, many algebraic polynomials (Hassani et al., 2013; Alamian et al., 2008; Farahani, 2013 ) have been introduced whose differentiation or integration or composition of both, obtained at a fixed point give topological indices. The M-polynomial (Deutsch and Klavzar, 2015; Kwun et al., 2017) is the most general polynomial to produce a large number of degree based topological indices. Rapid advances are created on a day-to-day basis to design fresh indices. Recently, some researchers put their attention on the neighborhood degree sum based indices (Hosamani, 2017; Mondal et al., 2021; Mondal et al., 2019; Mondal et al., 2021; Ghorbani and Hosseinzadeh, 2010; Ghorbani and Hosseinzadeh, 2013; Kulli, 2019; Kulli, 2019 ). To make the computation of these type of indices easier, present authors (Mondal et al., 2021; Verma et al., 2019) introduced the neighborhood M-polynomial (NM) whose function for neighborhood degree sum based indices is analogous to the function of the M-polynomial for degree based indices. Kwun et al. (2017) computed M-Polynomials and degree based topological indices of V-phenylenic nanotubes and nanotori. Present authors (Mondal et al., 2019) derived topological indices for praline graph of some graceful structures using M-polynomial. M-polynomial-based topological descriptors of chemical crystal structures and their applications were discussed in (Chu et al., 2020). M-polynomials and topological indices of linear chains of benzene, napthalene and anthracene were reported in (Li et al., 2020). Cancan et al. (2020) computed topological indices of silicate network via M-polynomial. M-polynomials and degree-based topological indices of the molecule copper (I) oxide were derived in (Chaudhry et al., 2021). Liu et al. (2019) obtained topological indices of nano-tubes via M-polynomial. Closed formulas of M-polynomial and topological descriptors of boron triangular nanotube were calculated in (Shin et al., 2020). M-polynomials of some nano-structures and zigzag edge coronoid fused by starphene were derived in (Raza et al., 2020; Afzal et al., 2020). Topological descriptors for the crystal structure of titanium difluoride ${TiF}_{2}$ were obtained in (Liu et al., 2018). Neighborhood degree sum based indices of molecular graphs were computed in (Mondal et al., 2021; Verma et al., 2019) using NM-polynomial. The M-polynomial and NM-polynomial of some anti-COVID-19 chemicals were evaluated in (Mondal et al., 2020).

One of the most rapidly and actively growing areas of modern science is the production of nanostructured materials of various geometric shapes, including nanowires, nanotubes, etc. The fascination with nanostructured materials stems from their physicochemical, structural, magnetic, and conductive properties, which differ significantly from bulk samples of similar composition. Kozlovskiy and Zdorovets (2020) presented the results of a study of the structural properties and phase composition of $Co / {CoCo}_{2} O_{4}$ nanowires obtained by electrochemical deposition into the pores of template matrices based on polyethylene terephthalate. Study of the effect of thermal annealing on the change in the structural properties and phase composition of metal Co nanostructures, as well as their use as a basis for lithium-ion batteries were reported in (Zdorovets and Kozlovskiy, 2019). Green synthesis for producing nanoparticles (NP) is a biological method which is safe, economical, eco-friendly and reduces the use of chemicals. Synthesis of Ag silver NP using an aqueous extract of Acorus Calamus rhizome was reported in (Sudhakar et al., 2015). To evaluate the antibacterial, antioxidant, and antitumor effects of obtained Ag silver NP, Nakkala et al. (2014) performed synthesis involving an aqueous extract of rhizome Acorus Calamus. Green synthesis of barium ferrite nanoparticles using rhizome extract of Acorus Calamus was reported in (Thakur et al., 2020). The synthesis of ceramic nanostructured materials with multifunctional characteristics is an important area of research related to the production and study of new materials. Zhumatayeva et al. (2020) reported the outcomes of a study of structural characteristics, as well as the possibility of using ${Li}_{0.15} {Sr}_{0.85} {TiO}_{3}$ ceramics as anode materials for lithium-ion batteries. Critical behavior of ${La}_{0.825} {Sr}_{0.175} {MnO}_{2.912}$ anion-deficient manganite in the magnetic phase transition region was analysed in (Trukhanov et al., 2007). Synthesis and structure of nanocrystalline manganite ${La}_{0.50} {Ba}_{0.50} {MnO}_{3}$ was described in (Trukhanov et al., 2008). Nano-structures based on iron-nickel compounds have found their application in biomedical applications, such as carriers for targeted drug delivery, bases for lithium-ion batteries, etc. Structures based on iron-cobalt compounds and their oxide forms are increasingly used as magnetic sensors. Kozlovskiy et al. (2020) reported the outcomes of the synthesis and subsequent phase transformations of FeCo nanowires depending on the annealing temperature. High chemical stability and corrosion resistance, in combination with functional magnetic properties of ${BaFe}_{12} O_{19}$ M-type hexaferrites or BaM opens up a wide range of possibilities for potential applications. Recently, the strong correlation of the heat treatment conditions (annealing), crystal structure parameters, microstructure and magnetic properties evolution in BaM nanohexaferrites is presented in (Trukhanov et al., 2021). Titanium is the ninth most plentiful metal in the Earth’s crust. $TiO 2$ , the most common titanium chemical, has diverse uses from anti-corrosion, self-cleaning lubricants and paintings to solar panels and photocatalysts (Fujishima et al., 1999). The semi-conductive characteristics of ${TiO}_{2}$ nanotubes lead in a powerful ionic interaction between the support and an electrode, enhancing catalytic efficiency in redox rections. The ${TiO}_{2}$ nanotube is also used as electrode of dye-sensitized solar cells. Titanium difluoride is a water-insoluble form of titanium to use for oxygen-sensitive implementations like metal processing. Fluoride materials have a variety of applications in present technologies and research, from petroleum refining and coating to synthetic chemistry and pharmaceutical manufacturing. To design a nanotube and nanoparticle with the suggested characteristics, structurally delicate properties such as fracture strength and yield stress can be controlled. The topological index is a numeric measure of molecule that describes the topology of molecular graph under testing in several ways. That is why, researchers obtained different topological indices (Liu et al., 2018; Munir et al., 2016; De, 2016; Liu et al., 2016 ) for the aforesaid structures. In this work, our focus is on titanium based structures. We aim to obtain some neighborhood degree sum based indices of titania nanotube ${TiO}_{2}$ and crystallographic structure of ${TiF}_{2}$ via neighborhood M-polynomial approach.

The rest of the manuscript is constructed as follows. Section 2 contains some preliminaries required to obtain the main findings. Section 3 contains the methodology required to obtain the main results. Section 4 is divided into two subsections: first one deals with the computation of titania nanotube ${TiO}_{2}$ and the later contains the computation for crystallographic structure of ${TiF}_{2}$ . The outcomes are compartively studied in Section 5. The work is concluded in Section 6.

2 Preliminaries

Topological indices based on degrees play a major role in the field of chemical graph theory among several classes of topological indices. Gutman and Trinajstić (1972) introduced the first degree-based molecular descriptor in 1972 which nowadays is known as Zagreb index. In 1975, Randić (1975) invented the branching index to characterize the molecular branching that was later renamed as connectivity index. Nowadays, most authors refer to it as to the Randić index. The degree based indices based on degree of end nodes of edges for a graph $ϒ$ are defined as follows: $I (ϒ) = \sum_{uv \in E (ϒ)} F (d_{u}, d_{v}),$ where $F (d_{u}, d_{v})$ is defined for different well-established descriptors in Table 1.

Table 1 Formulation of degree based topological indices.

$I (ϒ)$	$F (d_{u}, d_{v})$	$I (ϒ)$	$F (d_{u}, d_{v})$
First Zagreb index ( $M_{1} (ϒ)$ )(Gutman and Trinajstić, 1972)	$d_{u} + d_{v}$	Second Zagreb index ( $M_{2} (ϒ)$ )(Gutman and Trinajstić, 1972)	$d_{u} d_{v}$
Randić index ( $R (ϒ)$ )(Randić, 1975)	$\frac{1}{\sqrt{d_{u} d_{v}}}$	Forgotten topological index ( $F (ϒ)$ )(Furtula and Gutman, 2015)	$d_{u}^{2} + d_{v}^{2}$
Inverse sum indeg index ( $ISI (ϒ)$ ) (Vukicević and Gašperov, 2010)	$\frac{d_{u} d_{v}}{d_{u} + d_{v}}$	Redefined third Zagreb index ( ${ReZG}_{3} (ϒ)$ ) (Ranjini et al., 2013)	$d_{u} d_{v} (d_{u} + d_{v})$

For more degree based indices, readers are referred to (Nagesh and Girish, 2020; De, 2018; De, 2018; Numan et al., 2021; De, 2019 ). The M-polynomial of a graph $ϒ$ is defined as,

(1)

M (ϒ; x, y) = \sum_{i ⩽ j} m_{(i, j)} x^{i} y^{j} .

Where

m_{(i, j)}

is the total count of edges uv

\in

E (ϒ)

such that

\{d_{u}, d_{v}\} = \{i, j\}

. The neighborhood M-polynomial of a graph

ϒ

is defined as,

(2)

NM (ϒ; x, y) = \sum_{i ⩽ j} χ_{(i, j)} x^{i} y^{j} .

Where

χ_{(i, j)}

is the total count of edges uv

\in

E (ϒ)

such that

\{δ_{u}, δ_{v}\} = \{i, j\}

. We use

NM (ϒ)

for

NM (ϒ; x, y)

in this article. Now we set an example to describe aforesaid definitions. From Fig. 1, we have

d_{a} = 1, d_{b} = 2, d_{c} = 2, d_{d} = 1

δ_{a} = 2, δ_{b} = 3, δ_{c} = 3, δ_{d} = 2

. Thus, we have

m_{(1, 2)} = 2, m_{(2, 2)} = 1, χ_{(2, 3)} = 2, χ_{(3, 3)} = 1

. Therefore from Eqs. (1), (2), we have for n-butane,

M (G) = 2 {xy}^{2} + x^{2} y^{2}, NM (G) = 2 x^{2} y^{3} + x^{3} y^{3}

. Neighborhood degree sum based topological indices defined on edge collection of a graph

ϒ

can be represented as

I (ϒ) = \sum_{uv \in E (ϒ)} f (δ_{u}, δ_{v}),

where

f (δ_{u}, δ_{v})

is the function of

δ_{u}, δ_{v}

used in Table 2 to formulate neighborhood degree sum based indices. The above formula can also be written as

(3)

I (ϒ) = \sum_{i ⩽ j} m_{(i, j)} f (i, j) .

Formulations of neighborhood degree sum based indices and their relations with the NM-polynomial are shown in Table 2.

Table 2 Formulation of topological indices using NM-polynomial for a graph

ϒ

Topological Index	f(x,y)	Derivation from $NM (ϒ)$ .
Third version of Zagreb index ( $M_{1}^{'} (ϒ)$ ) (Ghorbani and Hosseinzadeh, 2010)	$x + y$	$(D_{x} + D_{y}) (NM (ϒ))$ at $x = y = 1$
Neighborhood second Zagreb index ( $M_{2}^{*} (ϒ)$ )(Mondal et al., 2019)	xy	$(D_{x} D_{y}) (NM (ϒ))$ at $x = y = 1$
Neighborhood forgotten topological index ( $F_{N}^{*} (ϒ)$ ) (Mondal et al., 2019)	$x^{2} + y^{2}$	$(D_{x}^{2} + D_{y}^{2}) (NM (ϒ))$ at $x = y = 1$
Neighborhood second modified Zagreb index( $^{nm} M_{2} (ϒ)$ ) (Mondal et al., 2021)	$\frac{1}{xy}$	$(S_{x} S_{y}) (NM (ϒ))$ at $x = y = 1$
Neighborhood general Randić index( ${NR}_{α} (ϒ)$ ) (Mondal et al., 2021)	${(xy)}^{α}$	$(D_{x}^{α} D_{y}^{α}) (NM (ϒ))$ at $x = y = 1$
Third NDe index ( ${ND}_{3} (ϒ)$ ) (Mondal et al., 2021)	$xy (x + y)$	$D_{x} D_{y} (D_{x} + D_{y}) (NM (ϒ))$ at $x = y = 1$
[1ex] Fifth NDe index ( ${ND}_{5} (ϒ)$ ) (Mondal et al., 2021)	$\frac{x^{2} + y^{2}}{xy}$	$(D_{x} S_{y} + S_{x} D_{y}) (NM (ϒ))$ at $x = y = 1$
Neighborhood Harmonic index ( $NH (ϒ)$ ) (Mondal et al., 2021)	$\frac{2}{x + y}$	$2 S_{x} J (NM (ϒ))$ at $x = 1$
Neighborhood inverse sum index ( $NI (ϒ)$ ) (Mondal et al., 2021)	$\frac{xy}{x + y}$	$S_{x} {JD}_{x} D_{y} (NM (ϒ))$ at $x = 1$
Sanskruti index ( $S (ϒ)$ ) (Hosamani, 2017)	$\frac{xy}{{(x + y - 2)}^{3}}$	$S_{x}^{3} Q_{- 2} {JD}_{x}^{3} D_{y}^{3} (NM (ϒ))$ at $x = 1$

Here, $NM (ϒ)$ is a function of $x, y$ and $D_{x} (NM (ϒ)) = x \frac{\partial (NM (ϒ))}{\partial x}, D_{y} (NM (ϒ)) = y \frac{\partial (NM (ϒ))}{\partial y},$ $S_{x} (NM (ϒ)) = \int_{0}^{x} \frac{(NM (ϒ)) |_{x = z}}{z} dz, S_{y} (NM (ϒ)) = \int_{0}^{y} \frac{(NM (ϒ)) |_{y = z}}{z} dz,$ $J (NM (ϒ)) = (NM (ϒ)) |_{y = x}, Q_{α} (NM (ϒ)) = x^{α} NM (ϒ) .$ For $α = - \frac{1}{2}$ in ${NR}_{α}$ , we have the fourth NDe index (Mondal et al., 2021) as follows: ${ND}_{4} (ϒ) = \sum_{uv \in E (ϒ)} \frac{1}{\sqrt{δ_{u} δ_{v}}} .$

3 Methodology

The present work deals with some neighborhood degree sum based indices of two Titanium compounds: ${TiO}_{2}$ and ${TiF}_{2}$ . First of all, the NM-polynomials of both the networks are computed and then using some calculus operators, different neighborhood degree sum based indices are recovered. We utilize combinatorial computation, graph theoretical tools and edge partition method to obtain the outcomes. The graphical representations of the results are shown using Maple 2015. A comparative study of the findings is performed via 3d line plotting in Matlab 2017.

4 Main results

In this section, we give our key analytical results and divide the section into two subsections.

4.1

4.1 Computational aspects of Titania nanotubes ${TiO}_{2} [p, q]$

In this section we consider the titania nanotube ${TiO}_{2} [p, q]$ . The structure of the Titania nanotube ${TiO}_{2} [p, q]$ is depicted in Fig. 2. We compute the NM-polynomial of ${TiO}_{2} [p, q]$ -nanotube in the following theorem.

Theorem 1

Let $ϒ$ be the ${TiO}_{2} [p, q]$ -nanotube. Then we have, $\begin{matrix} NM (ϒ) = & 2 x^{5} y^{7} + 2 x^{5} y^{10} + 2 {qx}^{7} y^{9} + 2 {qx}^{7} y^{13} + 4 {qx}^{8} y^{9} + 2 {qx}^{9} y^{10} + 6 {px}^{9} y^{11} \\ + 3 {px}^{9} y^{13} + 2 q (2 p + 1) x^{10} y^{13} + 2 q (p - 1) x^{11} y^{13} + 2 q (3 p - 2) x^{13} y^{13} . \end{matrix}$

Proof

The ${TiO}_{2} [p, q]$ -nanotube has $12 pq + 9 p + 6 q + 4$ number of edges. Its edge set can be partitioned as follows: $∣ E_{\{5, 7\}} ∣ = ∣ \{uv \in E (ϒ) : δ_{u} = 5, δ_{v} = 7\} ∣ = 2 = χ_{(5, 7)},$ $∣ E_{\{5, 10\}} ∣ = ∣ \{uv \in E (ϒ) : δ_{u} = 5, δ_{v} = 10\} ∣ = 2 = χ_{(5, 10)},$ $∣ E_{\{7, 9\}} ∣ = ∣ \{uv \in E (ϒ) : δ_{u} = 7, δ_{v} = 9\} ∣ = 2 q = χ_{(7, 9)},$ $∣ E_{\{7, 13\}} ∣ = ∣ \{uv \in E (ϒ) : δ_{u} = 7, δ_{v} = 13\} ∣ = 2 q = χ_{(7, 13)},$ $∣ E_{\{8, 9\}} ∣ = ∣ \{uv \in E (ϒ) : δ_{u} = 8, δ_{v} = 9\} ∣ = 4 q = χ_{(8, 9)},$ $∣ E_{\{9, 10\}} ∣ = ∣ \{uv \in E (ϒ) : δ_{u} = 9, δ_{v} = 10\} ∣ = 2 q = χ_{(9, 10)},$ $∣ E_{\{9, 11\}} ∣ = ∣ \{uv \in E (ϒ) : δ_{u} = 9, δ_{v} = 11\} ∣ = 6 p = χ_{(9, 11)},$ $∣ E_{\{9, 13\}} ∣ = ∣ \{uv \in E (ϒ) : δ_{u} = 9, δ_{v} = 13\} ∣ = 3 p = χ_{(9, 13)},$ $∣ E_{\{10, 13\}} ∣ = ∣ \{uv \in E (ϒ) : δ_{u} = 10, δ_{v} = 13\} ∣ = 2 q (2 p + 1) = χ_{(10, 13)},$ $∣ E_{\{11, 13\}} ∣ = ∣ \{uv \in E (ϒ) : δ_{u} = 11, δ_{v} = 13\} ∣ = 2 q (p - 1) = χ_{(11, 13)},$ $∣ E_{\{13, 13\}} ∣ = ∣ \{uv \in E (ϒ) : δ_{u} = 13, δ_{v} = 13\} ∣ = 2 q (3 p - 2) = χ_{(13, 13)} .$ From the definition, the NM-polynomial of $ϒ$ is obtained as follows. $\begin{matrix} NM (ϒ) = \sum_{i ⩽ j} χ_{(i, j)} x^{i} y^{j} \\ = m_{(5, 7)} x^{5} y^{7} + m_{(5, 10)} x^{5} y^{10} + m_{(7, 9)} x^{7} y^{9} + m_{(7, 13)} x^{7} y^{13} \\ + m_{(8, 9)} x^{8} y^{9} + m_{(9, 10)} x^{9} y^{10} + m_{(9, 11)} x^{9} y^{11} + m_{(9, 13)} x^{9} y^{13} \\ + m_{(10, 13)} x^{10} y^{13} + m_{(11, 13)} x^{11} y^{13} + m_{(13, 13)} x^{13} y^{13} . \\ = 2 x^{5} y^{7} + 2 x^{5} y^{10} + 2 {qx}^{7} y^{9} + 2 {qx}^{7} y^{13} + 4 {qx}^{8} y^{9} + 2 {qx}^{9} y^{10} + 6 {px}^{9} y^{11} \\ + 3 {px}^{9} y^{13} + 2 q (2 p + 1) x^{10} y^{13} + 2 q (p - 1) x^{11} y^{13} + 2 q (3 p - 2) x^{13} y^{13} . \end{matrix}$ This completes the proof. □

The structure of TiO 2 [ p , q ] -nanotubes, for p = 6 and q = 4 .

Now using this NM-polynomial, we calculate some neighborhood degree sum based indices of the ${TiO}_{2} [p, q]$ -nanotube as follows.

Theorem 2

Let $ϒ$ be the ${TiO}_{2} [p, q]$ -nanotube. Then we have,

$M_{1}^{'} (ϒ) = 296 pq + 186 p + 72 q + 54$ ,
$M_{2}^{*} (ϒ) = 1820 pq + 945 p + 74 q + 170$ ,
$F_{N}^{*} (ϒ) = 3684 pq + 1962 p + 244 q + 398$ ,
$^{nm} M_{2} (ϒ) = 0.08 pq + 0.086 p + 0.109 q + 0.097$ ,
${NR}_{α} (ϒ) = [4 {(130)}^{α} + 2 {(143)}^{α} + 6 {(13)}^{2 α}] pq + [6 {(99)}^{α} + 3 {(117)}^{α}] p + [2 {(63)}^{α} + 2 {(91)}^{α} + 4 {(72)}^{α} + 2 {(90)}^{α} + 2 {(130)}^{α} - 2 {(143)}^{α} - 4 {(13)}^{2 α}] q + 2 [{(35)}^{α} + {(50)}^{α}]$ ,
${ND}_{3} (ϒ) = 45188 pq + 19602 p - 4488 q + 2340$ ,
${ND}_{5} (ϒ) = 24.333 pq + 18.653 p + 13.078 q + 9.228$ ,
$NH (ϒ) = 0.976 pq + 0.873 p + 0.831 q + 0.6$ ,
$NI (ϒ) = 73.525 pq + 45.654 p + 16.778 q + 12.5$ ,
$S (ϒ) = 3593.145 pq + 1598.855 p - 512.809 q + 199.542$ .

Proof

Let $NM (ϒ) = f (x, y) = 2 x^{5} y^{7} + 2 x^{5} y^{10} + 2 {qx}^{7} y^{9} + 2 {qx}^{7} y^{13} + 4 {qx}^{8} y^{9} + 2 {qx}^{9} y^{10} + 6 {px}^{9} y^{11} + 3 {px}^{9} y^{13} + 2 q (2 p + 1) x^{10} y^{13} + 2 q (p - 1) x^{11} y^{13} + 2 q (3 p - 2) x^{13} y^{13}$ .

Then we have, $\begin{matrix} (D_{x} + D_{y}) (f (x, y)) = & 24 x^{5} y^{7} + 30 x^{5} y^{10} + 32 {qx}^{7} y^{9} + 40 {qx}^{7} y^{13} + 68 {qx}^{8} y^{9} + \\ 38 {qx}^{9} y^{10} + 120 {px}^{9} y^{11} + 66 {px}^{9} y^{13} + 46 q (2 p + 1) x^{10} y^{13} \\ + 48 q (p - 1) x^{11} y^{13} + 52 q (3 p - 2) x^{13} y^{13}, \\ D_{x} D_{y} (f (x, y)) = & 70 x^{5} y^{7} + 100 x^{5} y^{10} + 126 {qx}^{7} y^{9} + 182 {qx}^{7} y^{13} + \\ 288 {qx}^{8} y^{9} + 180 {qx}^{9} y^{10} + 594 {px}^{9} y^{11} + 351 {px}^{9} y^{13} + \\ 260 q (2 p + 1) x^{10} y^{13} + 286 q (p - 1) x^{11} y^{13} \\ + 338 q (3 p - 2) x^{13} y^{13}, \\ (D_{x}^{2} + D_{y}^{2}) (f (x, y)) = & 148 x^{5} y^{7} + 250 x^{5} y^{10} + 260 {qx}^{7} y^{9} + 436 {qx}^{7} y^{13} + \\ 580 {qx}^{8} y^{9} + 362 {qx}^{9} y^{10} + 1212 {px}^{9} y^{11} + 750 {px}^{9} y^{13} + \\ 538 q (2 p + 1) x^{10} y^{13} + 580 q (p - 1) x^{11} y^{13} + 676 q (3 p \\ - 2) x^{13} y^{13}, \\ S_{x} S_{y} (f (x, y)) = & \frac{2}{35} x^{5} y^{7} + \frac{1}{25} x^{5} y^{10} + \frac{2}{63} {qx}^{7} y^{9} + \frac{2}{91} {qx}^{7} y^{13} + \frac{1}{18} {qx}^{8} y^{9} \\ + \frac{1}{45} {qx}^{9} y^{10} + \frac{2}{33} {px}^{9} y^{11} + \frac{1}{39} {px}^{9} y^{13} + \frac{1}{65} q (2 p + \\ 1) x^{10} y^{13} + \frac{2}{143} q (p - 1) x^{11} y^{13} + \frac{2}{169} q (3 p - 2) x^{13} y^{13}, \\ D_{x}^{α} D_{y}^{α} (f (x, y)) = & 2 {(35)}^{α} x^{5} y^{7} + 2 {(50)}^{α} x^{5} y^{10} + 2 q {(63)}^{α} x^{7} y^{9} + \\ 2 q {(91)}^{α} x^{7} y^{13} + 4 q {(72)}^{α} x^{8} y^{9} + 2 q {(90)}^{α} x^{9} y^{10} + \\ 6 p {(99)}^{α} x^{9} y^{11} + 3 p {(117)}^{α} x^{9} y^{13} + 2 q (2 p \\ + 1) {(130)}^{α} x^{10} y^{13} + 2 q (p - 1) {(143)}^{α} x^{11} y^{13} + \\ 2 q (3 p - 2) {(13)}^{2 α} x^{13} y^{13}, \\ D_{x} D_{y} (D_{x} + D_{y}) (f (x, y)) = & 840 x^{5} y^{7} + 1500 x^{5} y^{10} + 2016 {qx}^{7} y^{9} + 3640 {qx}^{7} y^{13} + \\ 4896 {qx}^{8} y^{9} + 3420 {qx}^{9} y^{10} + 11880 {px}^{9} y^{11} \\ + 7722 {px}^{9} y^{13} + 5980 q (2 p + 1) x^{10} y^{13} + 6864 q (p - \\ 1) x^{11} y^{13} + 8788 q (3 p - 2) x^{13} y^{13}, \\ (D_{x} S_{y} + S_{x} D_{y}) (f (x, y)) = & \frac{148}{35} x^{5} y^{7} + 5 x^{5} y^{10} + \frac{260}{63} {qx}^{7} y^{9} + \frac{436}{91} {qx}^{7} y^{13} + \\ \frac{145}{18} {qx}^{8} y^{9} + \frac{181}{45} {qx}^{9} y^{10} + \frac{404}{33} {px}^{9} y^{11} + \end{matrix}$ $\begin{matrix} \frac{250}{39} {px}^{9} y^{13} + \frac{269}{65} q (2 p + 1) x^{10} y^{13} + \frac{580}{143} q (p \\ - 1) x^{11} y^{13} + 4 q (3 p - 2) x^{13} y^{13}, \\ S_{x} J (f (x, y)) = & \frac{1}{6} x^{12} + \frac{2}{15} x^{15} + \frac{1}{8} {qx}^{16} + \frac{1}{10} {qx}^{20} + \frac{4}{17} {qx}^{17} + \frac{2}{19} {qx}^{19} \\ + \frac{3}{10} {px}^{20} + \frac{3}{22} {px}^{22} + \frac{2}{23} q (2 p + 1) x^{23} + \frac{1}{12} q (p \\ - 1) x^{24} + \frac{1}{13} q (3 p - 2) x^{26}, \\ S_{x} {JD}_{x} D_{y} (f (x, y)) = & \frac{35}{6} x^{12} + \frac{20}{3} x^{15} + \frac{63}{8} {qx}^{16} + \frac{91}{10} {qx}^{20} + \frac{288}{17} {qx}^{17} \\ + \frac{180}{19} {qx}^{19} + \frac{297}{10} {px}^{20} + \frac{351}{22} {px}^{22} + \frac{260}{23} q (2 p \\ + 1) x^{23} + \frac{143}{12} q (p - 1) x^{24} + 13 q (3 p - 2) x^{26}, \\ S_{x}^{3} Q_{- 2} {JD}_{x}^{3} D_{y}^{3} (f (x, y)) = & 85.75 x^{10} + 113.792 x^{13} + 182.25 {qx}^{14} + \\ 258.426 {qx}^{18} + 221.184 {qx}^{15} + 296.764 {qx}^{17} + \\ 998.25 {px}^{18} + 600.605 {px}^{20} + \\ 474.463 q (2 p + 1) x^{21} + 549.25 q (p - 1) x^{22} + \\ 698.323 q (3 p - 2) x^{24} . \end{matrix}$ Rest of the proof can be done easily using Table 2. □

The topological indices for ${TiO}_{2}$ [p,q] are depicted in Fig. 3, Fig. 4 and Fig. 5.

( a ) M 1 ′ , (b) M 2 ∗ , and (c) F N ∗ index of TiO 2 [p,q].

( a ) mn M 2 , (b) ND 3 , and (c) ND 5 index of TiO 2 [p,q].

( a ) NH, (b) NI, and (c) S index of TiO 2 [p,q].

Now putting $α = - \frac{1}{2}$ in Theorem 2, we obtain the following corollary.

Corollary 1

The fourth NDe index of ${TiO}_{2}$ [p,q] is given by ${ND}_{4} ({TiO}_{2} [p, q]) = 0.9796 pq + 0.8804 p + 0.84439 q + 0.621 .$

The vacancy defects, which are formed because of the absence of the atoms/ions from the lattice structure, are very common in all types of crystalline materials. Different properties of the titanium oxide-based nanomaterials are found to be explicitly dependent on the presence of various crystalline defects. Oxygen vacancies are the most frequent and foremost among them. Topological indices have two major subclasses: topostructural and topochemical. The former encode information strictly on molecular connectivity. The latter includes chemical features (atom and bond type) in addition to topological information. The present article deals with the topostructural indices. The argument of topostructural indices is always a molecular graph. By molecular graph of a chemical compound, we mean a simple connected graph whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds (irrespective of bond type) between them. Thus, it is apparent that oxygen vacancies on ${TiO}_{2}$ crystal reduces the number of nodes and edges of the molecular graph. As a result, the indices derived from Theorem 2 and Corollary 1, will alter. For instance, $M_{1}^{'}$ moves down for imposing oxygen vacancy on ${TiO}_{2}$ crystal. Each of the indices considered in this work can model different physico-chemical properties with powerful accuracy (Hosamani, 2017; Mondal et al., 2019; Mondal et al., 2021; Ghorbani and Hosseinzadeh, 2013; Mondal et al., 2020 ). It is therefore clear that oxygen vacancies play vital role in governing and even drastically changing various physico-chemical properties of titanium dioxide nanostructures.

4.2

4.2 Computational aspects of the crystallographic structure of ${TiF}_{2}$ [m, n, t].

Here we consider the crystal structure of Titanium difluoride ( ${TiF}_{2} [m, n, t]$ ). Its structure is shown in Fig. 6. The structure consists of $m \times n$ units in the plane and then stores them in t layers. We compute the NM-polynomial of the crystallographic structure of ${TiF}_{2}$ [m, n, t] as follows.

Theorem 3

Let $ϒ$ be the crystallographic structure ${TiF}_{2}$ [m, n, t]. Then we have, $\begin{matrix} NM (ϒ) = & 8 x^{4} y^{13} + 8 (m + n + t - 3) x^{8} y^{18} + 16 x^{13} y^{16} + 8 [2 mn + 2 mt + 2 nt - 2 m \\ + 2 n + 2 t + 1] x^{16} y^{18} + 8 [4 mnt - 2 (mn + mt + nt) + 1] x^{16} y^{24} + 8 (m + n + \\ t - 2) x^{18} y^{32} . \end{matrix}$

Proof

The ${TiF}_{2}$ [m, n, t] structure has $32 mnt$ number of edges. Its edge set has partitions as follows: $∣ E_{\{4, 13\}} ∣ = ∣ \{uv \in E (ϒ) : δ_{u} = 4, δ_{v} = 13\} ∣ = 8 = χ_{(4, 13)},$ $∣ E_{\{8, 18\}} ∣ = ∣ \{uv \in E (ϒ) : δ_{u} = 8, δ_{v} = 18\} ∣ = 8 (m + n + t - 3) = χ_{(8, 18)},$ $∣ E_{\{13, 16\}} ∣ = ∣ \{uv \in E (ϒ) : δ_{u} = 13, δ_{v} = 16\} ∣ = 16 = χ_{(13, 16)},$ $\begin{matrix} ∣ E_{\{16, 18\}} ∣ = & ∣ \{uv \in E (ϒ) : δ_{u} = 16, δ_{v} = 18\} ∣ = 8 [2 mn + 2 mt + 2 nt) - 2 m \\ + 2 n + 2 t + 1] = χ_{(16, 18)}, \end{matrix}$ $\begin{matrix} ∣ E_{\{16, 24\}} ∣ = & ∣ \{uv \in E (ϒ) : δ_{u} = 16, δ_{v} = 24\} ∣ = 8 [4 mnt - 2 mn - 2 mt - 2 nt + 1] \\ = χ_{(16, 24)}, \end{matrix}$ $∣ E_{\{18, 32\}} ∣ = ∣ \{uv \in E (ϒ) : δ_{u} = 18, δ_{v} = 32\} ∣ = 8 (m + n + t - 2) = χ_{(18, 32)} .$ From the definition, the NM-polynomial of G can be obtained like previous. Hence the proof. □

(a) Unit cell of TiF 2 [m,n,t] (b) Crystallographic structure of TiF 2 [4,1,2].

The surface representation of NM-polynomials for both the networks is shown in Fig. 7. The range of x and y axes is taken arbitrarily from −2 to 2, as except for the vertical axis values, no changes would appear in the figures for the higher or lower range. It shows the changes of the NM-polynomial for varying its arguments. Now using this NM-polynomial, we calculate neighborhood degree sum based indices of the ${TiF}_{2}$ [m, n, t] structure in the following theorem.

Theorem 4

Let $ϒ$ be ${TiF}_{2}$ [m, n, t]. Then we have,

$M_{1}^{'} (ϒ) = 1280 mnt - 96 mn - 96 mt - 96 nt + 64 m + 64 n + 64 t - 232$ ,
$M_{2}^{*} (ϒ) = 12288 mnt - 1536 mn - 1536 mt - 1536 nt + 1152 m + 1152 n + 1152 t - 3552$ ,
$F_{N}^{*} (ϒ) = 26624 mnt - 4032 mn - 4032 mt - 4032 nt + 4608 m + 4608 n + 4608 t - 11304$ ,
$^{nm} M_{2} (ϒ) = 0.083 mnt + 0.014 mn + 0.014 mt + 0.014 nt + 0.014 m + 0.014 n + 0.014 t + 0.085$ ,
${NR}_{α} (ϒ) = 8 {(52)}^{α} + 8 {(144)}^{α} (m + n + t - 3) + 16 {(208)}^{α} + 8 {(288)}^{α} [2 mn + 2 mt + 2 nt - 2 m - 2 n - 2 t + 1] + 8 {(384)}^{α} [4 mnt - 2 mn - 2 mt - 2 nt + 1] + 8 {(576)}^{α} (m + n + t - 2)$ ,
${ND}_{3} (ϒ) = 491520 mnt - 89088 mn - 89088 mt - 89088 nt + 103720 m + 103720 n + 103720 t - 245936$ ,
${ND}_{5} (ϒ) = 69.333 mnt - 2.444 mn - 2.444 mt - 2.444 nt + 8.056 m + 8.056 n + 8.056 t - 7.513$ ,
$NH (ϒ) = 1.6 mnt + 0.141 mn + 0.141 mt + 0.141 nt - 0.006 m - 0.006 n - 0.006 t + 0.429$ ,
$NI (ϒ) = 307.2 mnt - 18.07 mn - 18.07 mt - 18.07 nt + 0.938 m + 0.938 n + 0.938 t - 33.449$ ,
$S (ϒ) = 33021.2 mnt - 4846.6 mn - 4846.6 mt - 4846.6 nt + 3888 m + 3888 n + 3888 t - 11096.333$ .

Proof

Let $NM (ϒ) = f (x, y) = 8 x^{4} y^{13} + 8 (m + n + t - 3) x^{8} y^{18} + 16 x^{13} y^{16} + 8 [2 mn + 2 mt + 2 nt - 2 m - 2 n - 2 t + 1] x^{16} y^{18} + 8 [4 mnt - 2 mn - 2 mt - 2 nt + 1] x^{16} y^{24} + 8 (m + n + t - 2) x^{18} y^{32}$ .

Then we have, $\begin{matrix} (D_{x} + D_{y}) (f (x, y)) = & 136 x^{4} y^{13} + 208 (m + n + t - 3) x^{8} y^{18} + 464 x^{13} y^{16} + \\ 272 [2 mn + 2 mt + 2 nt - 2 m - 2 n - 2 t + 1] x^{16} y^{18} + \\ 320 [4 mnt - 2 mn - 2 mt - 2 nt + 1] x^{16} y^{24} + 400 (m + \\ n + t - 2) x^{18} y^{32}, \\ D_{x} D_{y} (f (x, y)) = & 416 x^{4} y^{13} + 1152 (m + n + t - 3) x^{8} y^{18} + 3328 x^{13} y^{16} + \\ 2304 [2 mn + 2 mt + 2 nt - 2 m - 2 n - 2 t + 1] x^{16} y^{18} + \\ 3072 [4 mnt - 2 mn - 2 mt - 2 nt + 1] x^{16} y^{24} + 4608 (m + \\ n + t - 2) x^{18} y^{32}, \\ (D_{x}^{2} + D_{y}^{2}) (f (x, y)) = & 1480 x^{4} y^{13} + 3104 (m + n + t - 3) x^{8} y^{18} + 6800 x^{13} y^{16} + \\ 4640 [2 mn + 2 mt + 2 nt - 2 m - 2 n - 2 t + 1] x^{16} y^{18} + \\ 6656 [4 mnt - 2 mn - 2 mt - 2 nt + 1] x^{16} y^{24} + 10784 (m + \\ n + t - 2) x^{18} y^{32}, \\ S_{x} S_{y} (f (x, y)) = & \frac{2}{13} x^{4} y^{13} + \frac{1}{18} (m + n + t - 3) x^{8} y^{18} + \frac{1}{13} x^{13} y^{16} + \\ \frac{1}{36} [2 mn + 2 mt + 2 nt - 2 m - 2 n - 2 t + 1] x^{16} y^{18} + \\ \frac{1}{48} [4 mnt - 2 mn - 2 mt - 2 nt + 1] x^{16} y^{24} + \frac{1}{72} (m + n + \\ t - 2) x^{18} y^{32}, \\ D_{x}^{α} D_{y}^{α} (f (x, y)) = & 8 {(52)}^{α} x^{4} y^{13} + 8 {(144)}^{α} (m + n + t - 3) x^{8} y^{18} + 16 {(208)}^{α} \\ x^{13} y^{16} + 8 {(288)}^{α} [2 mn + 2 mt + 2 nt - 2 m - 2 n - 2 t + 1] \\ x^{16} y^{18} + 8 {(384)}^{α} [4 mnt - 2 mn - 2 mt - 2 nt + 1] x^{16} y^{24} + \\ 8 {(576)}^{α} (m + n + t - 2) x^{18} y^{32}, \\ D_{x} D_{y} (D_{x} + D_{y}) (f (x, y)) = & 7072 x^{4} y^{13} + 29952 (m + n + t - 3) x^{8} y^{18} + 96512 x^{13} y^{16} \\ + 78336 [2 mn + 2 mt + 2 nt - 2 m - 2 n - 2 t + 1] x^{16} y^{18} + \\ 122880 [4 mnt - 2 mn - 2 mt - 2 nt + 1] x^{16} y^{24} + \\ 230440 (m + n + t - 2) x^{18} y^{32}, \\ (D_{x} S_{y} + S_{x} D_{y}) (f (x, y)) = & \frac{370}{13} x^{4} y^{13} + \frac{194}{9} (m + n + t - 3) x^{8} y^{18} + \frac{425}{13} x^{13} y^{16} + \\ \frac{145}{9} [2 mn + 2 mt + 2 nt - 2 m - 2 n - 2 t + 1] x^{16} y^{18} + \\ \frac{52}{3} [4 mnt - 2 mn - 2 mt - 2 nt + 1] x^{16} y^{24} + \frac{337}{18} (m + n \end{matrix}$ $\begin{matrix} + t - 2) x^{18} y^{32}, \\ S_{x} J (f (x, y)) = & \frac{8}{17} x^{17} + \frac{4}{13} (m + n + t - 3) x^{26} + \frac{16}{29} x^{29} + \frac{4}{17} [2 (mn + \\ mt + nt) - 2 (m + n + t) + 1] x^{34} + \frac{1}{5} [4 mnt - 2 mn - 2 mt \\ - 2 nt + 1] x^{40} + \frac{4}{25} (m + n + t - 2) x^{50}, \\ S_{x} {JD}_{x} D_{y} (f (x, y)) = & \frac{416}{17} x^{17} + \frac{576}{13} (m + n + t - 3) x^{26} + \frac{3328}{29} x^{29} + \\ \frac{1152}{17} [2 mn + 2 mt + 2 nt - 2 m - 2 n - 2 t + 1] x^{34} + \\ \frac{384}{5} [4 mnt - 2 mn - 2 mt - 2 nt + 1] x^{40} + \frac{2304}{25} (m + n \\ + t - 2) x^{50}, \\ S_{x}^{3} Q_{- 2} {JD}_{x}^{3} D_{y}^{3} (f (x, y)) = & 333.293 x^{15} + 1728 (m + n + t - 3) x^{24} + 7315.074 x^{27} + \\ 5832 [2 mn + 2 mt + 2 nt - 2 m - 2 n - 2 t + 1] x^{32} \\ + 8255.3 [4 mnt - 2 mn - 2 mt - 2 nt + 1] x^{38} + \\ 13824 (m + n + t - 2) x^{48} . \end{matrix}$ From Table 2, we can easily derive the required result. □

The NM-polynomal of (a) the TiO 2 [ 2 , 2 ] -nanotube and (b) the crystallographic structure TiF 2 [ 2 , 2 , 2 ] .

The topological indices for ${TiF}_{2}$ [m, n, 1] are depicted in Fig. 8, Fig. 9 and Fig. 10.

( a ) M 1 ′ , (b) M 2 ∗ , and (c) F N ∗ index of TiF 2 [m, n, 1].

( a ) mn M 2 , (b) ND 3 , and (c) ND 5 index of TiF 2 [m, n, 1].

( a ) NH, (b) NI, and (c) S index of TiF 2 [m, n, 1].

Now putting $α = - \frac{1}{2}$ in Theorem 2, we obtain the following corollary.

Corollary 2

The fourth NDe index of ${TiF}_{2}$ [m, n, t] is given by ${ND}_{4} ({TiF}_{2} [m, n, t]) = 0.9796 pq + 0.8804 p + 0.84439 q + 0.621 .$

5 Comparative study

Firstly, we consider comparative study among different neighborhood degree sum based indices for the considered structures, later some well-known degree based indices reported in Table 1 are compared with present findings.

Different neighborhood degree sum based indices for ${TiO}_{2}$ [p,q] and ${TiF}_{2}$ [m, n, 1] are compared graphically in Fig. 11 and Fig. 12, respectively. From the Figs. 3,4,5,8,9,10,11,12,13,14,15,16 , following remarks can be drawn. In each compound under consideration, all indices respond diversely. In case of each index, the structures have the following order: T( ${TiO}_{2}$ [p,q]) $<$ T( ${TiF}_{2}$ [m, n, t]), T denotes the topological index. From the vertical axes of the figures, we can conclude that for both the structures, the neighborhood degree sum based indices have the following order: ${ND}_{3} ⩾ F_{N}^{*} ⩾ M_{2}^{*} ⩾ M_{1}^{'} ⩾ {ND}_{5} ⩾ NI ⩾ {ND}_{4} ⩾ NH ⩾^{nm} M_{2}$ . The ${ND}_{3}$ has the most dominating nature compared to other neighborhood degree sum based indices, whereas $^{m} {NM}_{1}$ grew slowly. The indices computed in this paper increase as well as graph parameters increase.

Comparison of different topological indices for TiO 2 [p, q].

Comparison of different topological indices for TiF 2 [m, n, 1].

Comparison of M 1 ′ , M 2 ∗ and F N ∗ indices with corresponding degree based indices for TiO 2 [p, q].

Comparison of ND 3 , NI and ND 4 indices with corresponding degree based indices for TiO 2 [p, q].

Comparison of M 1 ′ , M 2 ∗ and F N ∗ indices with corresponding degree based indices for TiF 2 [m, n, 1].

Comparison of ND 3 , NI and ND 4 indices with corresponding degree based indices for TiF 2 [m, n, 1].

Many of the neighborhood degree sum based indices have similar formulations to degree based indices. Here we compare the behavior of the neighborhood degree sum based indices with their corresponding degree based indices for both the Titanium compounds through graphical representations. To plot degree based indices, explicit expressions of $M_{1}, M_{2}, R, F, ISI$ , and ${ReZG}_{3}$ for ${TiO}_{2}$ [p,q] and ${TiF}_{2}$ [m,n,t] are collected from (Munir et al., 2016; Liu et al., 2018; De, 2016; Liu et al., 2016 ). From Figs. 13–16 , one can say that neighborhood degree sum based descriptors show dominating nature over degree based descriptors except R and ${ND}_{4}$ . From Figs. 14, 16, it is clear to state that $R ⩾ {ND}_{4}$ . There is therefore a sizable variation of the descriptors for the structures investigated.

6 Remarks and conclusions

In this article, we have studied the topological properties of the ${TiO}_{2}$ nanotube and the crystallographic structure of ${TiF}_{2}$ . Firstly, the general form of neighborhood M-polynomial for the structures are obtained and then using that polynomials, some neighborhood degree sum based indices are recovered. To visualize the results, their graphical representations also have been made. Each of the indices considered in this work can model different physico-chemical properties with powerful accuracy. The correlation coefficients (r) of $M_{1}^{'}, M_{2}^{*}, F_{N}^{*},^{nm} M_{2}, {ND}_{3}, NH$ , and NI with entropy and acentric factor for octane isomers are $⩾ 0.93$ . Surprisingly, correlation coefficient of NI with acentric factor is 0.99. The correlation of ${ND}_{5}$ with different physico-chemical properties for alkanes is also notable ( $r ⩾ 0.9$ ). Besides the structure–property/structure–activity modelling, a good descriptor should discriminate isomers. The isomer discrimination ability of neighborhood degree sum based indices is remarkable in comparison with degree based indices. Thus the considered indices are useful to describe the structural features of molecular graph. The computation of such indices of aforesaid useful structures is therefore worth-while and the findings can be helpful to understand the topology of the structures.

Declaration

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Availability of data and material

Not applicable.

Funding

This work was supported by the United Arab Emirates University [Award Number: G00003271].

Authors’ contributions

SM- Conceptualization, Writing- Original draft preparation, Software; ND- Investigation, Writing- Reviewing and Editing; MI-Writing-Original draft preparation; AP- Supervision, Validation.

Acknowledgments

The first author is obliged to the Department of Science and Technology (DST), Government of India for the Inspire Fellowship [IF170148].

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Introduction

Preliminaries

Methodology

Main results

Computational aspects of Titania nanotubes TiO 2 [ p , q ]
Computational aspects of the crystallographic structure of TiF 2 [m, n, t].

Comparative study

Remarks and conclusions

Declaration

Consent for publication

Availability of data and material

Funding

Authors’ contributions

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[Google Scholar]

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[37] Nakkala J.R., Mata R., Gupta A.K., Sadras S.R., . Biological activities of green silver nanoparticles synthesized with acorous calamus rhizome extract. Eur. J. Med. Chem.. 2014;85:784-794.
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[42] Shin D.Y., Hussain S., Afzal F., Park C., Afzal D., Farahani M.R., . Closed Formulas for Some New Degree Based Topological Descriptors Using M-polynomial and Boron Triangular Nanotube. Front Chem.. 2020;8:613873.
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[Google Scholar]

[44] Thakur A., Sharma N., Bhatti M., Sharma M., Trukhanov A.V., Trukhanov S.V., Panina L.V., Astapovich K.A., Thakur P., . Synthesis of barium ferrite nanoparticles using rhizome extract of Acorus Calamus: Characterization and its efficacy against different plant phytopathogenic fungi. Nano-Struct. Nano-Objects.. 2020;24:100599.
[Google Scholar]

[45] Trukhanov S.V., Trukhanov A.V., Vasil’ev A.N., Maignan A., Szymczak H., . Critical behavior of La_0.825Sr_0.175MnO_2.912 anion-deficient manganite in the magnetic phase transition region. JETP Lett.. 2007;85:507-512.
[Google Scholar]

[46] Trukhanov S.V., Fedotova V.V., Trukhanov A.V., Stepin S.G., Szymczak H., . Synthesis and structure of nanocrystalline La_0.50Ba_0.50MnO₃. Crystallogr. Rep.. 2008;53:1177-1180.
[Google Scholar]

[47] Trukhanov A.V., Darwish K.A., Salem M.M., Hemeda O.M., Abdel A.M.I., Darwish M.A., Kaniukov E.Y., Podgornaya S.V., Turchenko V.A., Tishkevich D.I., Zubar T.I., Astapovich K.A., Kostishyn V.G., Trukhanov S.V., . Impact of the heat treatment conditions on crystal structre, morphology and magnetic properties evolution in BaM nanohexaferrites. J. Alloys Compd.. 2021;866:158961.
[Google Scholar]

[48] Verma A., Mondal S., De N., Pal A., . Topological Properties of Bismuth tri-iodide Using Neighborhood M-Polynomial. International Journal of Mathematics Trends and Technology. 2019;65(10):83-90.
[Google Scholar]

[49] Vukicević D., Gašperov M., . Bond additive modeling 1. Adriatic indices. Croat. Chem. Acta.. 2010;83:243-260.
[Google Scholar]

[50] Zdorovets M.V., Kozlovskiy A.L., . Investigation of phase transformations and corrosion resistance in Co/CoCo2O4 nanowires and their potential use as a basis for lithium-ion batteries. Sci. Rep.. 2019;9:16646.
[Google Scholar]

[51] Zhumatayeva I.Z., Kenzhina I.E., Kozlovskiy A.L., Zdorovets M.V., . The study of the prospects for the use of Li0.15Sr0.85TiO3 ceramics. J. Mater. Sci.: Mater. Electron.. 2020;31:6764-6772.
[Google Scholar]

Neighborhood M-polynomial of titanium compounds

Abstract

Keywords

Graph

Topological index

Neighborhood M-polynomial

Titania nanotube TiO2

Crystallographic structure TiF2

Primary: 05C35

Secondary: 05C07

05C40

1 Introduction

2 Preliminaries

3 Methodology

4 Main results

4.1 Computational aspects of Titania nanotubes TiO 2 [ p , q ]

4.2 Computational aspects of the crystallographic structure of TiF 2 [m, n, t].

5 Comparative study

6 Remarks and conclusions

Declaration

Consent for publication

Availability of data and material

Funding

Authors’ contributions

Acknowledgments

Declaration of Competing Interest

References

4.1 Computational aspects of Titania nanotubes ${TiO}_{2} [p, q]$

4.2 Computational aspects of the crystallographic structure of ${TiF}_{2}$ [m, n, t].