Research Article  Open Access
Muhammad Tanveer Hussain, Muhammad Javaid, Usman Ali, Ali Raza, Md Nur Alam, "Comparing Zinc Oxide and Zinc SilicateRelated MetalOrganic Networks via ConnectionBased Zagreb Indices", Journal of Chemistry, vol. 2021, Article ID 5066394, 16 pages, 2021. https://doi.org/10.1155/2021/5066394
Comparing Zinc Oxide and Zinc SilicateRelated MetalOrganic Networks via ConnectionBased Zagreb Indices
Abstract
Metalorganic 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). These 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 zincrelated MONs. The versatile applications of these MONs make them helpful tools in many fields of science in recent decade. In this paper, we discuss the two different zinc oxide and zinc silicate related MONs according to the number of increasing layers of metal and organic ligands together. We also compute the connectionbased 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 zincrelated MONs by using numerical values of connectionbased Zagreb indices. Finally, we conclude that zinc silicaterelated MON is better than zinc oxiderelated MON for all values of n.
1. Introduction
There are so many chemical compounds in the field of chemistry. One of the most popular recent chemical compounds is metalorganic network (MON) which consists of metal ions and organic ligands. A new MON with zinc as the metal ion and benzene1,3dicarboxylic acid as the organic ligand (linker) has been synthesized with the help of hydrothermal method. This MON is also used as a selective nanoadsorbent for the preconcentration and extraction of trace amounts of cadmium with the help of solidphase 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]. Znrelated MONs as chemical sensors could be converted into devices for luminescent characteristics [3]. The electronrich Tconjugated fluorescent ligands are friendly to make ZnMONs with nucleophilic properties in efficient luminescent sensors [4].
The low toxicity of zinc ions as the desirable character is considered to introduce Znrelated MONs into bioapplication domains, especially drug carries. The antibacterial activity has been validated by combining different antibiotic drugs and metals [5]. In reality, is an endogenous lowtoxic transition metal cation which is widely used in dermatology as a cicatrizing agent and skin moisturizer with astringent, antidandruff, antibacterial, and antiinflammatory agents [6]. In nonlinear optically active MONs, is commonly used as a connecting point to prevail undesired dd transitions in the visible region. Moreover, the toxicology, biomedical applications, and their biocompatibility are recently reported production procedures of zincrelated MONs. For more details, refer to [7].
Eddaoudi et al. [8] synchronized the isoreticular series (IRMOF1 to IRMOF16) of 16 highly crystalline materials. The free and fixed diameter of pores from IRMOF1 to IRMOF16 varies in the range of 3.8–19.1 and 12.8–28.8, respectively. The design of an IRMOF10 series based on MON5 was initiated by determining the reaction conditions necessary to produce the octahedral cluster with a ditopic linear carboxylate. Therefore, many IRMOF structures can be developed by using zinc oxide octahedral clusters as the metal corners linked via diverse organic dicarboxylate linkers and different threedimensional cubic networks are formed. For more information, see [9]. All the IRMOFs have the expected topology of and happened through the prototype IRMOF1 in which an oxidecentered tetrahedron is edge bridged. Some IRMOFs such as IRMOF8, IRMOF10, IRMOF12, and IRMOF16 have been seen in noncrystalline porous systems for xerogels and aerogels (16). For further investigation, see [7, 10–12].
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 MONrelated 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 [20–22]. First, distancebased topological index (TI) was discovered by Wiener to study the different properties of chemical compounds (boiling point of paraffin) in 1947 [23]. The very wellreputed firstdegreebased TI was discovered by Gutman and Trinajstić to check the chemical physibility on the total πelectron 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 fourlayered neural networks and MONs [26, 27]. Ali et al. [28] computed connectionbased indices and coindices for the product of molecular networks. Gutman and Furtula discussed various topological properties of different molecular structures; see [29–31]. Ali and Trinajstic [32] and Javaid et al. [33] computed different connectionbased TIs of graphs under different operations. Moreover, a variety of networks has been defined with the help of connection numberbased TIs [34–37].
In this paper, we compute the connectionbased Zagreb indices of two different zincrelated MONs such as zinc oxide (ZNOX (n) = IRMOF10) and zinc silicate (ZNCL (n) = IRMOF14) networks with respect to the increasing layers, , taking both metal nodes and linkers together. The 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.
2. Preliminaries
The 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. The distance between two vertices p and q is the shortest path between them. It is denoted by . In general [37], is the open mneighborhood set of q, where m represents a positive integer and is called mdistance degree of a vertex q. In particular,(i)If , = degree of q (number of vertices at distance one from q)(ii)If , = connection 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 , second Zagreb index , and third Zagreb index are defined as follows:(a)(b)(c)These degreebased TIs are defined by Gutman and Trinajstic [24]. These are abundantly used to predict better findings in molecular networks such as ZE isomerism, absolute value of correlation coefficient, entropy, acentric factor, and heat capacity.
Definition 2. For a (molecular) network G, the first ZCI , second ZCI , and modified first ZCI are defined as follows:(a)(b)(c)These connectionbased TIs are defined by Ali and Trinajstic [32] (2018). They 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 and modified third Zagreb connection index are defined as follows:(a)(b)
Definition 4. For a (molecular) network G, the modified fourth Zagreb connection index is defined as follows:These connectionbased TIs are defined by Javaid et al. [35] to compute the exact solutions of several wheelrelated 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. The 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 and hydrogen gas. It also reduces those metal ions whose reduction potentials are higher than . Zinc oxide is mostly used in making rubber, enamels, glazes, pigment in white paint, photoconductive surfaces, and protective coating for other metals. Zinc oxiderelated MON is , which is also known as IRMOF10. IRMOF9 is a catenated version of IRMOF10. IRMOF10 is threedimensional cubic structures with a pore size diameter of 16.7/20.2 . In Figure 1, the zinc oxiderelated 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 is the most charming class of minerals. Silicate is the chemical mixture of metal oxide or metal carbonate with sand. The basic unit of silicate is tetrahedron. So, all silicates gain tetrahedron. In chemistry, oxygen ions and silicon ions are represented by the corner vertices and centre vertices of , respectively. In graph theory, we represent corner vertices and centre vertices of 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 silicaterelated MON is , which is also known as IRMOF14. IRMOF14 is threedimensional cubic structures with a pore size diameter of 14.7/20.1 . In Figure 2, the zinc silicaterelated 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. Then, , where equality holds if and only if G is a free network.
Lemma 2. (see [35]). Let G be a connected and free network with u vertices and e edges. Then,(i)(ii)
Lemma 3. (see [25]). Let G be a connected and free network with u vertices and e edges. Also, . Then(i) if (ii) if
3. 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 oxiderelated MON (ZNOX (n)). Let be the zinc oxide network of dimension n in the plane, see Figure 1. The partitions of with respect to the vertex set and edge set are and . We can easily see each vertex of degrees 2, 3, and 4. We have , , and , where V_{1} = 42n +30, V_{2} = 26n + 14, and V_{3} = 2n + 2. So, = = V_{1} + V_{2} + V_{3} = 70n + 46. Now, the partitions of vertices according to connection number are V_{1} = { ∈ V (H), = 2}, V_{2} = { ∈ V (H) = 3}, V_{3} = { ∈ V (H) = 4}, V_{4} = { ∈ V (H) = 5}, and V_{5} = { ∈ V (H) = 8}, where V_{1} = 2n + 6, V_{2} = 28n+20, V_{3} = 30n + 10, V_{4} = 8n + 8, and V_{5} = 2n + 2. So, V (H) = = V_{1} + V_{2} + V_{3} + V_{4} + V_{5} = 70n + 46. These vertex partitions are shown in Tables 1 and 2.


There are four types of partitions of edge sets of H according to the degree as E (H) =  +  +  +  = 85n + 55, and there are seven types of partitions of edge sets of according to the connection number of vertices as E (H) =  +  +  +  +  +  +  = 85n + 55. These edge partitions are shown in Tables 3 and 4.


Theorem 1. Let be a zinc oxide network of dimensions n ≥ 3. Then, the first Zagreb connection index is
Proof. By definition,
Theorem 2. Let be a zinc oxide network of dimensions n ≥ 3. Then, the second Zagreb connection index is
Proof. By definition,
Theorem 3. Let be a zinc oxide network of dimensions n ≥ 3. Then, the modified first Zagreb connection index is
Proof. By definition,
Theorem 4. Let be a zinc oxide network of dimensions n ≥ 3. Then, the modified second Zagreb connection index is
Proof. By definition,
Theorem 5. Let be a zinc oxide network of dimensions n ≥ 3. Then, the modified third Zagreb connection index is
Proof. By definition,
Theorem 6. Let be a zinc oxide network of dimensions n ≥ 3. Then, the modified fourth Zagreb connection index is
Proof. By definition,
4. Main Results Based on Zinc Silicate Network (ZNSL (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 silicaterelated MON (ZNSL (n)). Let be the zinc silicate network of dimension n in the plane, see Figure 2. The partitions of with respect to the vertex set and edge set are and . We can easily see each vertex of degrees 2, 3, and 4. We have V_{1} = { ∈ V (K) = 2}, V_{2} = { ∈ V (K) = 3}, and V_{3} = { ∈ V (K) = 4}, where V_{1} = 42n + 30, V_{2} = 38n + 18, and V_{3} = 2n + 2. So, V (K) = = V_{1} + V_{2} + V_{3} = 82n + 50. Now, the partitions of vertices according to connection number are V_{1} = { ∈ V (K) = 2}, V_{2} = { ∈ V (K) = 3}, V_{3} = { ∈ V (K) = 4}, V_{4} = { ∈ V (K) = 5}, V_{5} = { ∈ V (K) = 6}, and V_{6} = { ∈ V (K) = 8}, where V_{1} = 2n+6, V_{2} = 16n+16, V_{3} = 48n+16, V_{4} = 8n+8, V_{5} = 6n + 2, and V_{6} = 2n + 2. So, V (K) = = V_{1} + V_{2} + V_{3} + V_{4} + V_{5} = 82n + 50. These vertex partitions are shown in Tables 5 and 6.


There are four types of partitions of edge sets of according to the degree as E (K) =  +  +  +  = 103n + 61, and there are seven types of partitions of edge sets of according to the connection number of vertices as E (K) =  +  +  +  +  +  +  +  +  = 103n + 61. These edge partitions are shown in Tables 7 and 8.


Theorem 7. Let be a zinc silicate network of dimensions n ≥ 3. Then, the first Zagreb connection index is
Proof. By definition,
Theorem 8. Let be a zinc silicate network of dimensions n ≥ 3. Then, the second Zagreb connection index is
Proof. By definition,
Theorem 9. Let be a zinc silicate network of dimensions n ≥ 3. Then, the modified first Zagreb connection index is
Proof. By definition,
Theorem 10. Let be a zinc silicate network of dimensions n ≥ 3. Then, the modified second Zagreb connection index is
Proof. By definition,