Comparing Zinc Oxide- and Zinc Silicate-Related Metal-Organic Networks via Connection-Based Zagreb Indices

Metal-organic networks (MONs) are among the unique complex and porous chemical compounds. So, these chemical compounds consist of metal ions (vertices) and organic ligands (edges between vertices). +ese networks represent large pore volume, extreme surface area, morphology, excellent chemical stability, highly porous and crystalline materials, and octahedral clusters. MONs are mostly used in assessment of chemicals, gas and energy storage devices, sensing, separation and purification of different gases, heterogeneous catalysis, environmental hazard, toxicology, adsorption analysis, biomedical applications, and biocompatibility. Recently, drug delivery, cancer imaging, and biosensing have been investigated by biomedical applications of zincrelatedMONs.+e versatile applications of theseMONsmake them helpful tools in many fields of science in recent decade. In this paper, we discuss the two different zinc oxide and zinc silicate relatedMONs according to the number of increasing layers of metal and organic ligands together. We also compute the connection-based Zagreb indices such as first Zagreb connection index (ZCI), second ZCI, modified first ZCI, modified second ZCI, modified third ZCI, and modified fourth ZCI. Moreover, a comparison is also included between the zinc-relatedMONs by using numerical values of connection-based Zagreb indices. Finally, we conclude that zinc silicate-related MON is better than zinc oxide-related MON for all values of n.


Introduction
ere are so many chemical compounds in the field of chemistry. One of the most popular recent chemical compounds is metal-organic network (MON) which consists of metal ions and organic ligands. A new MON with zinc as the metal ion and benzene-1,3-dicarboxylic acid as the organic ligand (linker) has been synthesized with the help of hydrothermal method. is MON is also used as a selective nanoadsorbent for the preconcentration and extraction of trace amounts of cadmium with the help of solid-phase extraction method. A category of crystalline and porous materials is porous coordination polymers which are newly known as MONs [1]. One of the most important aspects which can be considered in the matter of MONs in bioapplications is their surface modification and control of their particle size distribution [2]. Zn-related MONs as chemical sensors could be converted into devices for luminescent characteristics [3]. e electron-rich T-conjugated fluorescent ligands are friendly to make Zn-MONs with nucleophilic properties in efficient luminescent sensors [4]. e low toxicity of zinc ions as the desirable character is considered to introduce Zn-related MONs into bioapplication domains, especially drug carries. e antibacterial activity has been validated by combining different antibiotic drugs and metals [5]. In reality, Zn 2+ is an endogenous low-toxic transition metal cation which is widely used in dermatology as a cicatrizing agent and skin moisturizer with astringent, antidandruff, antibacterial, and anti-inflammatory agents [6]. In nonlinear optically active MONs, Zn 2+ is commonly used as a connecting point to prevail undesired d-d transitions in the visible region. Moreover, the toxicology, biomedical applications, and their biocompatibility are recently reported production procedures of zinc-related MONs. For more details, refer to [7].
Eddaoudi et al. [8] synchronized the isoreticular series (IRMOF-1 to IRMOF-16) of 16 highly crystalline materials. e free and fixed diameter of pores from IRMOF-1 to IRMOF-16 varies in the range of 3.8-19.1 and 12.8-28.8, respectively. e design of an IRMOF-10 series based on MON-5 was initiated by determining the reaction conditions necessary to produce the octahedral cluster with a ditopic linear carboxylate. erefore, many IRMOF structures can be developed by using zinc oxide octahedral clusters (Zn 4 O(CO) 2 ) as the metal corners linked via diverse organic dicarboxylate linkers and different three-dimensional cubic networks are formed. For more information, see [9]. All the IRMOFs have the expected topology of CaB 6 (13) and happened through the prototype IRMOF-1 in which an oxide-centered Zn 4 O tetrahedron is edge bridged. Some IRMOFs such as IRMOF-8, IRMOF-10, IRMOF-12, and IRMOF-16 have been seen in noncrystalline porous systems for SiO 2 xerogels and aerogels (16). For further investigation, see [7,[10][11][12].
MONs also predict the physicochemical properties such as grafting active groups [13], impregnating suitable active material [14], ion exchange [15], preparing composites with different substances [11], changing organic ligands and postsynthetic ligands [16], and biosensors enhancing sensitivity, response time, and selectivity [17]. Yap et al. [18] and Lin et al. [19] presented the recent progress in precursors on the preparation of several nanostructures and MON-related applications such as sensing, photocatalysis, electrocatalysis, supercapacitors, catalyst for production of fine chemicals, and lithium ion batteries. Graph theory provides useful tools in the field of modern chemistry which represent the chemical and physical properties of chemical compounds such as heat of formation, heat of evaporation, flash point, melting point, boiling point, temperature, pressure, density, retention in chromatography, and tension and partition coefficient [20][21][22]. First, distance-based topological index (TI) was discovered by Wiener to study the different properties of chemical compounds (boiling point of paraffin) in 1947 [23]. e very well-reputed first-degreebased TI was discovered by Gutman and Trinajstić to check the chemical physibility on the total π-electron energy of the chemical compounds (alternant hydrocarbons) in 1972 [24].
Recently, Zhao et al. [25] introduced two connection number (number of vertices at distance two) based TIs to compute the general results for modified Zagreb connection indices of subdivision and semitotal point operations on graphs. Nowadays, these degree and connection numberbased TIs are abundantly used in the topological properties of four-layered neural networks and MONs [26,27]. Ali et al. [28] computed connection-based indices and coindices for the product of molecular networks. Gutman and Furtula discussed various topological properties of different molecular structures; see [29][30][31]. Ali and Trinajstic [32] and Javaid et al. [33] computed different connection-based TIs of graphs under different operations. Moreover, a variety of networks has been defined with the help of connection number-based TIs [34][35][36][37].
In this paper, we compute the connection-based Zagreb indices of two different zinc-related MONs such as zinc oxide (ZNOX (n) � IRMOF-10) and zinc silicate (ZNCL (n) � IRMOF-14) networks with respect to the increasing layers, n ≥ 3, taking both metal nodes and linkers together. e rest of the paper is organized as follows. Section 2 provides the preliminaries, definitions, and some important results which can be used in the main results. Sections 3 and 4 provides the main results for zinc oxide and zinc silicate networks, and Section 5 provides comparisons and conclusions.

Preliminaries
e vertex and edge sets are V (G) and E (G) for a simple and connected network G. |V (G)| and |E (G)| are the cardinalities of vertex set and edge set which are equal to u and e, respectively. In a connected network, there is a path between two vertices. e distance between two vertices p and q is the shortest path between them. It is denoted by d G (p, q). In general [37], number of q (number of vertices at distance two from q) For more terminologies and notations, see [36] and references therein. Definition 1. For a (molecular) network G, the first Zagreb index (M 1 (G)), second Zagreb index (M 2 (G)), and third Zagreb index (M 3 (G)) are defined as follows: ] ese connection-based TIs are defined by Ali and Trinajstic [32] (2018). ey also reported that the modified first Zagreb connection index has better correlation coefficient value for the thirteen physicochemical properties of octane isomers than classical Zagreb indices.
Definition 3. For a (molecular) network G, the modified second Zagreb connection index (ZC * 2 (G)) and modified third Zagreb connection index (ZC * 3 (G)) are defined as follows: Definition 4. For a (molecular) network G, the modified fourth Zagreb connection index (ZC * 4 (G)) is defined as follows: ese connection-based TIs are defined by Javaid et al. [35] to compute the exact solutions of several wheel-related graphs.
Definition 5. Zinc oxide network (ZNOX (n)): a chemical compound zinc oxide (ZnO) is insoluble in water which is an inorganic compound of white powder shape with 5.61 g/cm 3 density. e zinc oxide is heated with carbon (coke) that reduces to the metal vapor to condense the liquid from which the solid metal freezes.
Zinc is a reactive metal to produce zinc ion (Zn 2+ ) and hydrogen gas. It also reduces those metal ions whose reduction potentials are higher than Zn 2+ . Zinc oxide is mostly used in making rubber, enamels, glazes, pigment in white paint, photoconductive surfaces, and protective coating for other metals. Zinc oxide-related MON is Zn 4 O(BPDC) 3 , which is also known as IRMOF-10. IRMOF-9 is a catenated version of IRMOF-10. IRMOF-10 is three-dimensional cubic structures with a pore size diameter of 16.7/20.2 A 0 . In Figure 1, the zinc oxide-related MON of dimension 3 is presented. In general, the vertices and edges in ZNOX (n) of dimension n are 70n + 46 and 85n + 55, respectively. For more understanding, see Figure 1.
Definition 6. Zinc silicate network (ZNSL (n)): silicate (SiO 4 ) is the most charming class of minerals. Silicate is the chemical mixture of metal oxide or metal carbonate with sand. e basic unit of silicate is tetrahedron. So, all silicates gain SiO 4 tetrahedron. In chemistry, oxygen ions and silicon ions are represented by the corner vertices and centre vertices of SiO 4 , respectively. In graph theory, we represent corner vertices and centre vertices of SiO 4 with oxygen nodes and silicon nodes. If we require a variety of silicate networks, it is easy to change the arrangement of the tetrahedron silicate. Zinc silicate-related MON is Zn 4 O(PDC) 3 , which is also known as IRMOF-14. IRMOF-14 is three-dimensional cubic structures with a pore size diameter of 14.7/20.1 A 0 . In Figure 2, the zinc silicate-related MON of dimension 3 is presented. In general, the vertices and edges in ZNSL (n) of dimension n are 82n + 50 and 103n + 61, respectively. For more understanding, see Figure 2. Now, we present some important results which are used in the main results.

Lemma 1. Let G be a connected network with u vertices and e edges. en, τ
where equality holds if and only if G is a C 3 , C 4 -free network.

Main Results Based on Zinc Oxide Network (ZNOX (n))
In this section, we compute the main results for first Zagreb connection index (ZCI), second ZCI, modified first ZCI, modified second ZCI, modified third ZCI, and modified fourth ZCI of zinc oxide-related MON (ZNOX (n)). Let H � ZNOX(n) be the zinc oxide network of dimension n in the plane, see Figure 1. e partitions of H with respect to the vertex set and edge set are V(H) and E(H). We can easily see each vertex of degrees 2, 3, and 4. We have   Tables 3 and 4.

Theorem 1. Let H � ZNOX(n) be a zinc oxide network of dimensions n ≥ 3. en, the first Zagreb connection index is
Proof. By definition,

Journal of Chemistry
Proof. By definition, Proof. By definition,

Theorem 4. Let H � ZNOX(n) be a zinc oxide network of dimensions n ≥ 3. en, the modified second Zagreb connection index is
Proof. By definition, Proof. By definition,  Journal of Chemistry

Theorem 6. Let H � ZNOX(n) be a zinc oxide network of dimensions n ≥ 3. en, the modified fourth Zagreb connection index is
Proof. By definition,
□ Theorem 10. Let K � ZNSL(n) be a zinc silicate network of dimensions n ≥ 3. en, the modified second Zagreb connection index is