Резюме. The Zagreb connection indices are molecular descriptors (topological indices) which have recently been introduced by Ali & Trinajstić (2018). A novel/ old modification of the first Zagreb index, Mol. Inform. , to appear]. This paper is devoted to establishing general expressions for calculating the Zagreb connection indices of a well-known nanostructure, namely the Titania nanotube \(\mathrm{TiO}_{2}\).

Ключови думи: molecular structure descriptor; Zagreb connection indices; titania nanotube

Introduction

Chemical compounds are associated with physical and chemical properties, and some of these compounds have biological activities. Actually, in the search for new antibacterial and various other medicinal drugs, pharmaceutical companies synthesize and test annually millions of new chemical compounds, characterizing in detail those compounds that show some promise (Balaban, 2013). In 2017, the number of substances registered in the Chemical Abstracts Service databases reached 135 million. Testing for biological activity is expensive, therefore many theoretical methods have been devised for correlating structures with biological activities or physical-chemical properties. One of the simplest such methods involves molecular structure descriptors or topological indices (Balaban, 2013; Devillers & Balaban, 1999).

Mathematical chemistry deals with the study and development of mathematical models of chemical phenomena (Basak, 2013). Chemical graph theory is one of the branches of mathematical chemistry. In this branch, graphs are used to represent chemical compounds, in which vertices correspond to the atoms while edges represent the covalent bonds between atoms (Gutman & Polansky, 1986). A numerical quantity calculated from a molecular graph is said to be a molecular structure descriptor, or more precisely, a topological index if it is invariant under graph isomorphism (Devillers & Balaban, 1999). The Wiener index is the first topological index proposed by the chemist Wiener in 1947 to model the boiling point of hydrocarbons (Todeschini & Consonni, 2009). Later on, many other topological indices were devised to predict certain physicochemical properties of chemical compounds (Kier & Lowell, 1976).

Titanium is a chemical element discovered by a German Chemist Martin Heinrich Klaproth in 1791. Its symbol is \(Ti\) . Titanium dioxide has a chemical formula \(\mathrm{TiO}_{2}\), also known as titania, which is a naturally occurring oxide of titanium.

Radushkevich & Lukyanovich (1952) discovered the carbon nanotubes. The reference (Kasaga et al., 1999) is the first publication on Titania nanotube. After then titania became a highly studied compound and up till now thousands of publications have been appeared on this compound (Roy et al., 2011).

The distance between two vertices \(u, v \in V(H)\) of a (connected) molecular graph \(H\), is the length of the shortest path joining them. The number of vertices at distance 2 from a vertex \(u \in V(H)\) of a (connected) molecular graph \(H\), is known aspaper the. connection Certainly, the number connection of \(u u\) number, and it will of a vertex be denoted in \(H\) bymust be at least is \(\sigma(u)\) throughout and this at most \(n-2\).

The first Zagreb connection index \(Z C_{1}\), second Zagreb connection index \(Z C_{2}\) and modified first Zagreb connection index \(Z C_{1}^{*}\) for a (connected) molecular graph \(H\) are defined (Ali & Trinajstić, 2017) as:

\[ \begin{aligned} & Z C_{1}(H)=\sum_{u \in V(G)}(\sigma(u))^{2}, \\ & Z C_{2}(H)=\sum_{u v \in E(G)} \sigma(u) \sigma(v), \\ & Z C_{1}^{*}(H)=\sum_{u v \in E(G)}(\sigma(u)+\sigma(v)) . \end{aligned} \] In this paper, we find general expressions for calculating the first Zagreb connection index, second Zagreb connection index, and modified first Zagreb connection index of the Titania nanotube \(\mathrm{TiO}_{2}\) (see Fig. 1 for its chemical structure and Fig. 2 for its molecular graph).

Main results

Denote by G the molecular graph of the \(\mathrm{TiO}_{2}\)–nanotube (see Figs. 1 and 2). Clearly, the connection number of any vertex of the molecular graph \(G\) of the \(\mathrm{TiO}_{2}\)–nanotube is among the numbers \(3,4,5,6,7,9,10\). We partition the vertex set \(V(G)\) of the graph \(G\) into some disjoint sets. Let \((G)=V_{3} \cup V_{4} \cup V_{5} \cup V_{6} \cup V_{7} \cup V_{9} \cup V_{10}\), where \(\quad V_{3}=\{u \in V(G) \mid \sigma(u)=3\}, \quad V_{4}=\{u \in V(G) \mid \sigma(u)=4\}\), \(V_{5}=\{u \in V(G) \mid \sigma(u)=5\}\), \(V_{6}=\{u \in V(G) \mid \sigma(u)=6\}\),

\(V_{7}=\{u \in V(G) \mid \sigma(u)=7\}\), \(V_{9}=\{u \in V(G) \mid \sigma(u)=9\}\), \(V_{10}=\{u \in V(G) \mid \sigma(u)=10\}\). Cardinalities of these sets are given in Table 1.

Figure 1. Chemical structure of \(\mathrm{TiO}_{3}[m, k]\)-nanotube, where \(m\) denotes the number of octagons in a column and \(k\) denotes the number of octagons in a row

Now, the first Zagreb connection index of \(G\) is defined as:

This formula can be rewritten as:

\[ \begin{aligned} & Z C_{1}(G)= \sum_{u \in V_{3}}[\sigma(u)]^{2}+\sum_{u \in V_{4}}[\sigma(u)]^{2}+\sum_{u \in V_{5}}[\sigma(u)]^{2} \\ &+\sum_{u \in V_{6}}[\sigma(u)]^{2}+\sum_{u \in V_{7}}[\sigma(u)]^{2} \\ &+\sum_{u \in V_{9}}[\sigma(u)]^{2}+\sum_{u \in V_{10}}[\sigma(u)]^{2} \\ & Z C_{1}(G)=9\left|V_{3}\right|+16\left|V_{4}\right|+25\left|V_{5}\right|+36\left|V_{6}\right|+49\left|V_{7}\right|+81\left|V_{9}\right|+100\left|V_{10}\right| \end{aligned} \]

Substituting the values from Table 1, we get

\[ Z C_{1}(G)=348 m k+84 k . \]

If we denote the number of vertices of a (connected) graph \(H\) having connection number \(i\) by \(y_{i}(H)\), and \(\varphi_{i}\) is any non-negative real valued function depending on \(i\), then we can define a more general atom’s connection-number-based topological index: \[ A C N(H)=\sum_{0 \leq i \leq n-2} y_{i}(H) \cdot \varphi_{i} \]

where \(n\) is the total number of vertices in the graph \(H\). For the molecular graph \(G\) of the \(\mathrm{TiO}_{2}[m, k]\)– nanotube, we have

(1)\( \operatorname{ACN}(H)=2 k\left[\varphi_{3}+\varphi_{4}+\varphi_{6}+\varphi_{9}-\varphi_{10}\right]+2 m k\left[\varphi_{5}+\varphi_{7}+\varphi_{10}\right] \)

If we take \(\varphi_{i}=i^{2}\), then Eq. (1) gives

\[ \begin{aligned} & Z C_{1}(G)=2 k\left[3^{2}+4^{2}+6^{2}+9^{2}-10^{2}\right]+2 m k\left[5^{2}+7^{2}+10^{2}\right] \\ & +2 m k\left[5^{2}+7^{2}+10^{2}\right]=84 k+348 m k=84 k+348 m k \end{aligned} \] Now, we calculate other two Zagreb connection indices of the molecular graph \(G\) of the \(\mathrm{TiO}_{2}[m, k]\)– nanotube. For this, we partition the edge set \(E(G)\) into some disjoint sets. Let us take

\[ E_{q}=\{e=u v \in E(G) \mid \sigma(u)+\sigma(v)=q\} \]

and

\[ E_{s}^{*}=\{e=u v \in E(G) \mid \sigma(u) \cdot \sigma(v)=s\} \]

Or more precisely,

\[ \begin{aligned} & E_{8}=E_{15}^{*}=\{e=u v \in E(G) \mid \sigma(u)=3, \sigma(u)=5\} \\ & E_{9}=E_{18}^{*}=\{e=u v \in E(G) \mid \sigma(u)=3, \sigma(u)=6\} \\ & E_{10}=E_{24}^{*}=\{e=u v \in E(G) \mid \sigma(u)=4, \sigma(u)=6\} \\ & E_{11}=E_{28}^{*}=\{e=u v \in E(G) \mid \sigma(u)=4, \sigma(u)=7\} \\ & E_{12}=E_{27}^{*}=\{e=u v \in E(G) \mid \sigma(u)=3, \sigma(u)=9\} \\ & E_{12}=E_{35}^{*}=\{e=u v \in E(G) \mid \sigma(u)=5, \sigma(u)=7\} \\ & E_{13}=E_{36}^{*}=\{e=u v \in E(G) \mid \sigma(u)=4, \sigma(u)=9\} \\ & E_{14}=E_{40}^{*}=\{e=u v \in E(G) \mid \sigma(u)=4, \sigma(u)=10\} \\ & E_{14}=E_{45}^{*}=\{e=u v \in E(G) \mid \sigma(u)=5, \sigma(u)=9\} \\ & E_{15}=E_{50}^{*}=\{e=u v \in E(G) \mid \sigma(u)=5, \sigma(u)=10\} \end{aligned} \]

Cardinalities of these sets are given in Table 2.

For the Table 2, we remark that

\[ \left|E_{12}\right|=\left|E_{27}^{*} \cup E_{35}^{*}\right|=4 m k-2 k \] and \[ \left|E_{14}\right|=\left|E_{40}^{*} \cup E_{45}^{*}\right|=4 k \] It is clear that

\[ E(G)=E_{8} \cup E_{9} \cup E_{10} \cup E_{11} \cup E_{12} \cup E_{13} \cup E_{14} \cup E_{15} . \] By using the definition of the modified first Zagreb connection index and using

Table 2, we have \[ Z C_{1}^{*}(G)=138 m k+52 k \] Also, we have \[ E(G)=E_{15}^{*} \cup E_{18}^{*} \cup E_{24}^{*} \cup E_{27}^{*} \cup E_{28}^{*} \cup E_{35}^{*} \cup E_{36}^{*} \cup E_{40}^{*} \cup E_{45}^{*} \cup E_{50}^{*} \] By using the definition of the second Zagreb connection index and using Table 2, we have

\[ Z C_{2}(G)=440 m k+12 k \]

Following (Ali & Trinajstić, 2018), denote by \(y_{i, j}(H)\) the number of those edges of a (connected) graph \(H\) which connect the vertices having connection numbers \(i, j\), and denote by \(\varphi_{i, j} \varphi_{i, j}\) any non-negative real valued function depending on , then bond incident connection-number (BIC) indices are defined as

\[ B I C(H)=\sum_{0 \leq i, j \leq n-2} y_{i, j}(G) \cdot \varphi_{i, j} \]

For the molecular graph \(G\) of the \(\mathrm{TiO}_{2}[\mathrm{~m}, \mathrm{k}]\)– nanotube, we have \[ \begin{aligned} \operatorname{BIC}(G)=2 k & {\left[2 \varphi_{3,5}+\varphi_{3,6}+\varphi_{4,6}+\varphi_{3,9}+2 \varphi_{4,7}-2 \varphi_{5,7}+\varphi_{4,9}+\varphi_{4,10}+\varphi_{5,9}-4 \varphi_{5,10}\right] } \\ & +2 m k\left[2 \varphi_{5,7}+3 \varphi_{5,10}\right] . \end{aligned} \] The above general expression can also be used to find the modified first Zagreb connection index and second Zagreb connection index of the molecular graph \(GG\) of the \(TiO_2\) –nanotube.

REFERENCES

Ali, A & Trinajstić, N. (2018). A novel/old modification of the first Zagreb index, Mol. Inform. (to appear).

Balaban, A. T. (2013). Chemical graph theory and the Sherlock Holmes principle. HYLE: Int. J. Phil. Chem., 19, 107 – 134.

Devillers, J. & Balaban, A.T. (1999). Topological indices and related descriptors in QSAR and QSPR. Singapore: Gordon & Breach.

Basak, S.C. (2013). Philosophy of mathematical chemistry: a personal perspective. HYLE: Int. J. Phil. Chem. 19, 3 – 17.

Gutman, I. & Polansky. O.E. (1986). Mathematical concepts of organic chemistry. Berlin: Springer.

Kasuga, T., Hiramatsu, M., Hoson, A., Sekino, T. & Niihara, K. (1999). Titania nanotubes prepared by chemical processing. Adv. Mat., 11, 1307 – 1311.

Kier, L.B. & Lowell, H.H. (1976). Molecular connectivity in chemistry and drug research. New York: Academic Press.

Radushkevich, L.V. & Lukyanovich, V.M. (1952). On the structure of carbon formed by thermal decomposition oxides of carbon in iron touch. J. Phys. Chem. (Russia), 26, 88 – 95.

Roy, P., Berger, S. & Schmuki, P. (2011). \(\mathrm{TiO}_{2}\) nanotubes: synthesis and applications. Angew. Chem. Int., 50, 2904 – 2939.

Todeschini, R. & Consonni, V. (2009). Molecular descriptions for chemoinformatics. Weinheim: Wiley.