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

Authors: Muhammad Tanveer Hussain; Muhammad Javaid; Usman Ali; Ali Raza; Md Nur Alam
Journal: Journal of Chemistry
 (2021)

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## Abstract

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). 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 zinc-related 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 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-related MONs 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 ofn.


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## Body

## 1. Introduction

There 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. This 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]. The electron-rich T-conjugated fluorescent ligands are friendly to make Zn-MONs with nucleophilic properties in efficient luminescent sensors [4].The low toxicity of zinc ions as the desirable character is considered to introduce Zn-related MONs into bioapplication domains, especially drug carries. The antibacterial activity has been validated by combining different antibiotic drugs and metals [5]. In reality, Zn2+ 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, Zn2+ 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. The 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. The 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. Therefore, many IRMOF structures can be developed by using zinc oxide octahedral clusters Zn4OCO2 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 CaB613 and happened through the prototype IRMOF-1 in which an oxide-centered Zn4O 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 SiO2 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 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 [20–22]. 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]. The very well-reputed first-degree-based 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 number-based 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 [29–31]. 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 [34–37].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. 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 areV (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 dGp,q. In general [37], NGq/m=p∈VG;dp,q=m is the open m-neighborhood set of q, where m represents a positive integer and NGq/m=dGq/m is called m-distance degree of a vertex q. In particular,(i)
Ifm=1, dGq/1=NGq/1=dGq = degree of q (number of vertices at distance one from q)(ii)
Ifm=2, dGq/2=NGq/2=τGq = 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) networkG, the first Zagreb index M1G, second Zagreb index M2G, and third Zagreb index M3G are defined as follows:(a)
M1G=∑q∈VGdGq2=∑pq∈EGdGp+dGq(b)
M2G=∑pq∈EGdGp×dGq(c)
M3G=∑q∈VGdGq3=∑pq∈EGdG2p+dG2q
These degree-based 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) networkG, the first ZCI ZC1G, second ZCI ZC2G, and modified first ZCI ZC1∗G are defined as follows:(a)
ZC1G=∑q∈VGτGq2(b)
ZC2G=∑pq∈EGτGp×τGq(c)
ZC1∗G=∑pq∈EG=τGp+τGq
These connection-based 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) networkG, the modified second Zagreb connection index ZC2∗G and modified third Zagreb connection index ZC3∗G are defined as follows:(a)
ZC2∗G=∑pq∈EGdGpτGq+dGqτGp(b)
ZC3∗G=∑pq∈EGdGpτGp+dGqτGqDefinition 4.
For a (molecular) networkG, the modified fourth Zagreb connection index ZC4∗G is defined as follows:(1)ZC4∗G=∑pq∈EGdGpτGp×dGqτGq.
These 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/cm3 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.(2)ZnOs+Cs⟶Zng+COg
Zinc is a reactive metal to produce zinc ionZn2+ and hydrogen gas. It also reduces those metal ions whose reduction potentials are higher than Zn2+. 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 Zn4OBPDC3, 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 A0. 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.Figure 1
Zinc oxide networkZNOXn≅H for n = 3.Definition 6.
Zinc silicate network (ZNSL (n)): silicate SiO4 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 SiO4 tetrahedron. In chemistry, oxygen ions and silicon ions are represented by the corner vertices and centre vertices of SiO4, respectively. In graph theory, we represent corner vertices and centre vertices of SiO4 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 Zn4OPDC3, which is also known as IRMOF-14. IRMOF-14 is three-dimensional cubic structures with a pore size diameter of 14.7/20.1 A0. 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.Figure 2
Zinc silicate networkZNSLn≅K for n = 3.Lemma 1.
LetG be a connected network with u vertices and e edges. Then, τGp+dGp≤∑q∈NGpdGq, where equality holds if and only if G is a C3,C4-free network.Lemma 2.
(see [35]). Let G be a connected and C3,C4-free network with u vertices and e edges. Then,(i)
∑q∈VGdGq=2e(ii)
∑q∈VGτGq=M1G−2eLemma 3.
(see [25]). Let G be a connected and C3,C4-free network with u vertices and e edges. Also, G≅Pu. Then(i)
ZC2∗G=8u−22 if u≥4(ii)
ZC3∗G=8u−22 if u≥3


## 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 oxide-related MON (ZNOX (n)). Let H≅ZNOXn be the zinc oxide network of dimension n in the plane, see Figure 1. The partitions of H with respect to the vertex set and edge set are VH and EH. We can easily see each vertex of degrees 2, 3, and 4. We have V1=v∈VH|dv=2, V2=v∈VH|dv=3, and V3=v∈VH|dv=4, where |V1| = 42n +30, |V2| = 26n + 14, and |V3| = 2n + 2. So, VH = v = |V1| + |V2| + |V3| = 70n + 46. Now, the partitions of vertices according to connection number are V1 = {v ∈ V (H)|, τv = 2}, V2 = {v ∈ V (H)|τv = 3}, V3 = {v ∈ V (H)|τv = 4}, V4 = {v ∈ V (H)|τv = 5}, and V5 = {v ∈ V (H)|τv = 8}, where |V1| = 2n + 6, |V2| = 28n+20, |V3| = 30n + 10, |V4| = 8n + 8, and |V5| = 2n + 2. So, |V (H)| = v = |V1| + |V2| + |V3| + |V4| + |V5| = 70n + 46. These vertex partitions are shown in Tables 1 and 2.Table 1
Partitions ofH’s vertices according to the degree.
dv234dv42n + 3026n + 142n + 2Table 2
Partitions ofH’s vertices according to the connection number.
τv23458τv2n + 628n + 2030n + 108n + 82n + 2There are four types of partitions of edge sets ofH according to the degree as |E (H)| = |E2,2d| + |E2,3d| + |E3,3d| + |E3,4d| = 85n + 55, and there are seven types of partitions of edge sets of H according to the connection number of vertices as |E (H)| = |E2,3c| + |E3,3c| + |E3,4c| + |E3,5c| + |E4,4c| + |E4,5c| + |E5,8c| = 85n + 55. These edge partitions are shown in Tables 3 and 4.Table 3
Partitions ofH’s edges according to the degree.
Edp,dqdE2,2dE2,3dE3,3dE3,4dEdp,dqd6n + 1652n + 289n + 38n + 8Table 4
Partition ofH’s edges according to the connection number.
Eτp,τqcE2,3cE3,3cE3,4cE3,5cE4,4cE4,5cE5,8cEτp,τqc4n + 124n + 1224n + 124n + 1221n + 712n + 48n + 8Theorem 1.
LetH≅ZNOXn be a zinc oxide network of dimensions n ≥ 3. Then, the first Zagreb connection index is(3)ZC1H=1068n+692.Proof.
By definition,(4)ZC1G=∑q∈VGτGq2=2n+622+28n+2032+30n+1042+8n+852+2n+242=8n+24+252n+180+480n+160+200n+200+128n+128=1068n+692.Theorem 2.
LetH≅ZNOXn be a zinc oxide network of dimensions n ≥ 3. Then, the second Zagreb connection index is(5)ZC2H=1376n+896.Proof.
By definition,(6)ZC2G=∑pq∈EGτGp×τGq=∑pq∈E2,3cτHp×τHq+∑pq∈E3,3cτHp×τHq+∑pq∈E3,5cτHp×τHq+∑pq∈E4,5cτHp×τHq+∑pq∈E4,4cτHp×τHq+∑pq∈E3,4cτHp×τHq+∑pq∈E4,4cτHp×τHq+∑pq∈E5,8cτHp×τHq=E2,3H2×3+E3,3H3×3+E3,5H3×5+E4,5H4×5+E4,4H4×4+E3,4H3×4+E4,4H4×4+E5,8H5×8=4n+122×3+12n+43×3+4n+123×5+12n+44×5+12n+44×4+24n+83×4+9n+34×4+8n+85×8=24n+72+108n+36+60n+180+240n+80+192n+64+288n+96+144n+48+320n+320=1376n+896.Theorem 3.
LetH≅ZNOXn be a zinc oxide network of dimensions n ≥ 3. Then, the modified first Zagreb connection index is(7)ZC1∗H=672n+432.Proof.
By definition,(8)ZC1∗G=∑pq∈EGτGp+τGq=∑pq∈E2,3cτHp+τHq+∑pq∈E3,3cτHp×τHq+∑pq∈E3,5cτHp+τHq+∑pq∈E4,5cτHp×τHq+∑pq∈E4,4cτHp×τHq+∑pq∈E3,4cτHp×τHq+∑pq∈E4,4cτHp×τHq+∑pq∈E5,8cτHp×τHq=E2,3H2+3+E3,3H3+3+E3,5H3+5+E4,5H4+5+E4,4H4+4+E3,4H3+4+E4,4H4+4+E5,8H5+8=4n+125+12n+46+4n+128+12n+49+12n+48+24n+87+9n+38+8n+813=20n+60+72n+24+32n+96+108n+36+96n+32+168n+56+72n+24+104n+104=672n+432.Theorem 4.
LetH≅ZNOXn be a zinc oxide network of dimensions n ≥ 3. Then, the modified second Zagreb connection index is(9)ZC2∗H=1740n+1124.Proof.
By definition,(10)ZC2∗G=∑pq∈EGdGpτGq+dGqτGp=∑pq∈E2,3cdHpτHq+dHqτHp+∑pq∈E3,3cdHpτHq+dHqτHp+∑pq∈E3,5cdHpτHq+dHqτHp+∑pq∈E4,5cdHpτHq+dHqτHp+∑pq∈E4,4cdHpτHq+dHqτHp+∑pq∈E3,4cdHpτHq+dHqτHp+∑pq∈E4,4cdHpτHq+dHqτHp+∑pq∈E5,8cdHpτHq+dHqτHp=E2,3C2×3+2×2+E3,3C2×3+2×3+E3,5C2×5+3×3+E4,5C2×5+3×4+E4,4C2×4+3×4+E3,4C2×4+3×3+E4,4C3×4+3×4+E5,8C3×8+4×5=4n+1210+12n+412+4n+1219+12n+422+12n+420+24n+817+9n+324+8n+844=40n+120+144n+48+76n+228+264n+88+240n+80+408n+136+216n+72+352n+352=1740n+1124.Theorem 5.
LetH≅ZNOXn be a zinc oxide network of dimensions n ≥ 3. Then, the modified third Zagreb connection index is (11)ZC3∗H=1808n+1184.Proof.
By definition,(12)ZC3∗G=∑pq∈EGdGpτGp+dGqτGq=∑pq∈E2,3cdHpτHp+dHqτH\q+∑pq∈E3,3cdHpτHp+dHqτHq+∑pq∈E3,5cdHpτH\p+dHqτHq+∑pq∈E4,5cdHpτHp+dHqτHq+∑pq∈E4,4cdHpτHp+dHqτHq+∑pq∈E3,4cdHpτHp+dHqτHq+∑pq∈E4,4cdHpτHp+dHqτHq+∑pq∈E5,8cdHpτHp+dHqτHq=E2,3c2×2+2×3+E3,3c2×3+2×3+E3,5c2×3+3×5+E4,5c2×4+3×5+E4,4c2×4+3×4+E3,4c2×3+3×4+E4,4c∗3×4+3×4+E5,8c3×5+4×8=4n+1210+12n+412+4n+1221+12n+423+12n+420+24n+818+9n+324+8n+847=40n+120+144n+48+84n+252+276n+92+240n+80+432n+144+216n+72+376n+376=1808n+1184.Theorem 6.
LetH≅ZNOXn be a zinc oxide network of dimensions n ≥ 3. Then, the modified fourth Zagreb connection index is(13)ZC4∗H=10344n+7224.Proof.
By definition,(14)ZC4∗G=∑pq∈EGdGpτGq×dGqτGp=∑pq∈E2,3cdHpτHq×dHqτHp+∑pq∈E3,3cdHpτHq×dHqτHp+∑pq∈E3,5cdHpτHq×dHqτHp+∑pq∈E4,5cdHpτHq×dHqτHp+∑pq∈E4,4cdHpτHq×dHqτHp+∑pq∈E3,4cdHpτHq×dHqτHp+∑pq∈E4,4cdHpτHq×dHqτHp+∑pq∈E5,8cdHpτHq×dHqτHp=E2,3c2×3×2×2+E3,3c2×3×2×3+E3,5c2×5×3×3+E4,5c2×5×3×4+E4,4c2×4×3×4+E3,4c2×4×3×3+E4,4C∗3×4×3×4+E5,8C3×8×4×5=4n+1224+12n+436+4n+1290+12n+4120+12n+496+24n+872+9n+3144+8n+8480=96n+288+432n+144+360n+1080+1440n+480+1152n+384+1728n+576+1296n+432+3840n+3840=10344n+7224.


## 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 silicate-related MON (ZNSL (n)). Let K≅ZNSLn be the zinc silicate network of dimension n in the plane, see Figure 2. The partitions of K with respect to the vertex set and edge set are VK and EK. We can easily see each vertex of degrees 2, 3, and 4. We have V1 = {v ∈ V (K)|dv  = 2}, V2 = {v ∈ V (K)|dv = 3}, and V3 = {v ∈ V (K)|dv = 4}, where |V1| = 42n + 30, |V2| = 38n + 18, and |V3| = 2n + 2. So, |V (K)| = v = |V1| + |V2| + |V3| = 82n + 50. Now, the partitions of vertices according to connection number are V1 = {v ∈ V (K)|τv = 2}, V2 = {v ∈ V (K)|τv = 3}, V3 = {v ∈ V (K)|τv = 4}, V4 = {v ∈ V (K)|τv = 5}, V5 = {v ∈ V (K)|τv = 6}, and V6 = {v ∈ V (K)|τv = 8}, where |V1| = 2n+6, |V2| = 16n+16, |V3| = 48n+16, |V4| = 8n+8, |V5| = 6n + 2, and |V6| = 2n + 2. So, |V (K)| = v = |V1| + |V2| + |V3| + |V4| + |V5| = 82n + 50. These vertex partitions are shown in Tables 5 and 6.Table 5
Partitions ofK’s vertices according to the degree.
dv234dv42n + 3038n + 182n + 2Table 6
Partitions ofK’s vertices according to the connection number.
τv234568τv2n + 616n + 1648n + 168n + 86n + 22n + 2There are four types of partitions of edge sets ofK according to the degree as |E (K)| = |E2,2d| + |E2,3d| + |E3,3d| + |E3,4d| = 103n + 61, and there are seven types of partitions of edge sets of Kaccording to the connection number of vertices as |E (K)| = |E2,3c| + |E3,3c| + |E3,4c| + |E3,5c| + |E4,4c| + |E4,5c| + |E4,6c| + |E5,8c| + |E6,6c| = 103n + 61. These edge partitions are shown in Tables 7 and 8.Table 7
Partitions ofK’s edges according to the degree.
Edp,dqdE2,2dE2,3dE3,3dE3,4dEdp,dqd10n + 1464n + 3221n + 78n + 8Table 8
Partition ofK’s edges according to the connection number.
Eτp,τqcE2,3cE3,3cE3,4cE3,5cE4,4cE4,5cE4,6cE5,8cE6,6cEτp,τqc4n + 126n + 212n + 44n + 1242n + 1412n + 412n + 48n + 83n + 1Theorem 7.
LetK≅ZNSLn be a zinc silicate network of dimensions n ≥ 3. Then, the first Zagreb connection index is(15)ZC1K=1464n+824.Proof.
By definition,(16)ZC1G=∑q∈VGτGq2=2n+622+16n+1632+48n+1642+8n+852+6n+262+2n+282=8n+24+144n+144+768n+256+200n+200+216n+72+128n+128=1464n+824.Theorem 8.
LetK≅ZNSLn be a zinc silicate network of dimensions n ≥ 3. Then, the second Zagreb connection index is(17)ZC2K=1910n+1074.Proof.
By definition,(18)ZC2G=∑pq∈EGτGp×τGq=∑pq∈E2,3cτKp×τKq+∑pq∈E3,3cτKp×τKq+∑pq∈E3,5cτKp×τKq+∑pq∈E4,5cτKp×τKq+∑pq∈E4,4cτKp×τKq+∑pq∈E3,4cτKp×τKq+∑pq∈E4,4cτKp×τKq+∑pq∈E4,6cτKp×τKq+∑pq∈E6,6cτKp×τKq+∑pq∈E5,8cτKp×τKq=E2,3K2×3+E3,3K3×3+E3,5K3×5+E4,5K4×5+E4,4K4×4+E3,4K3×4+E4,4K4×4+E4,6K4×6+E6,6K6×6+E5,8K5×8=4n+122×3+6n+23×3+4n+123×5+12n+44×5+36n+124×4+12n+43×4+6n+24×4+12n+44×6+3n+16×6+8n+85×8=24n+72+54n+18+60n+180+240n+80+576n+192+144n+48+96n+32+288n+96+108n+36+320n+320=1910n+1074.Theorem 9.
LetK≅ZNSLn be a zinc silicate network of dimensions n ≥ 3. Then, the modified first Zagreb connection index is(19)ZC1∗K=876n+500.Proof.
By definition,(20)ZC1∗G=∑pq∈EGτGp+τGq=∑pq∈E2,3cτkp+τKq+∑pq∈E3,3cτKp+τKq+∑pq∈E3,5cτKp+τKq+∑pq∈E4,5cτKp+τKq+∑pq∈E4,4cτKp+τKq+∑pq∈E3,4cτKp+τKq+∑pq∈E4,4cτKp+τKq+∑pq∈E4,6cτKp+τKq+∑pq∈E6,6cτKp+τKq+∑pq∈E5,8cτKp+τKq=E2,3K2+3+E3,3K3+3+E3,5K3+5+E4,5K4+5+E4,4K4+4+E3,4K3+4+E4,4K4+4+E4,6K4+6+E6,6K6+6+E5,8K5+8=4n+122+3+6n+23+3+4n+123+5+12n+44+5+36n+124+4+12n+43+4+6n+24+4+12n+44+6+3n+16+6+8n+85+8=20n+60+36n+12+32n+96+108n+36+288n+96+84n+28+48n+16+120n+40+36n+12+104n+104=876n+500.Theorem 10.
LetK≅ZNSLn be a zinc silicate network of dimensions n ≥ 3. Then, the modified second Zagreb connection index is(21)ZC2∗K=2340n+1324.Proof.
By definition,(22)ZC2∗G=∑pq∈EGdGpτGq+dGqτGp=∑pq∈E2,3cdKpτKq+dKqτKp+∑pq∈E3,3cdKpτKq+dKqτKp+∑pq∈E3,5cdKpτKq+dKqτKp+∑pq∈E4,5cdKpτKq+dKqτKp+∑pq∈E4,4cdKpτKq+dKqτKp+∑pq∈E3,4cdKpτKq+dKqτKp+∑pq∈E4,4c∗dKpτKq+dKqτKp+∑pq∈E4,6cdKpτKq+dKqτKp+∑pq∈E6,6cdKpτKq+dKqτKp+∑pq∈E5,8cdKpτKq+dKqτKp=E2,3c2×3+2×2+E3,3c2×3+2×3+E3,5c2×5+3×3+E4,5c2×5+3×4+E4,4c2×4+3×4+E3,4c2×4+3×3+E4,4c∗3×4+3×4+E4,6c3×6+3×4+E6,6c3×6+3×6+E5,8c3×8+4×5=4n+1210+6n+212+4n+1219+12n+422+36n+1220+12n+417+6n+224+12n+430+3n+136+8n+844=40n+120+72n+24+76n+228+264n+88+720n+240+204n+68+144n+48+360n+120+108n+36+352n+352=2340n+1324.Theorem 11.
LetK≅ZNSLn be a zinc silicate network of dimensions n ≥ 3. Then, the modified third Zagreb connection index is(23)ZC3∗K=2396n+1380.Proof.
By definition,(24)ZC3∗G=∑pq∈EGdGpτGp+dGqτGq=∑pq∈E2,3cdKpτKp+dKqτKq+∑pq∈E3,3cdKpτKp+dKqτKq+∑pq∈E3,5cdKpτKp+dKqτKq+∑pq∈E4,5cdKpτKp+dKqτKq+∑pq∈E4,4cdKpτKp+dKqτKq+∑pq∈E3,4cdKpτKp+dKqτKq+∑pq∈E4,4c∗dKpτKp+dKqτKq+∑pq∈E4,6cdKpτKp+dKqτKq+∑pq∈E6,6cdKpτKp+dKqτKq+∑pq∈E5,8cdKpτKp+dKqτKq=E2,3c2×2+2×3+E3,3c2×3+2×3+E3,5c2×3+3×5+E4,5c2×4+3×5+E4,4c2×4+3×4+E3,4c2×3+3×4+E4,4c∗3×4+3×4+E4,6c3×4+3×6+E6,6c3×6+3×6+E5,8c3×5+4×8=4n+1210+6n+212+4n+1221+12n+423+36n+1220+12n+418+6n+224+12n+430+3n+136+8n+847=40n+120+72n+24+84n+252+276n+92+720n+240+216n+72+144n+48+360n+120+108n+36+376n+376=2396n+1380.Theorem 12.
LetK≅ZNSLn be a zinc silicate network of dimensions n ≥ 3. Then, the modified fourth Zagreb connection index is(25)ZC4∗K=14700n+8676.Proof.
By definition,(26)ZC4∗G=∑pq∈EGdGpτGq×dGqτGp=∑pq∈E2,3cdKpτKq×dKqτKp+∑pq∈E3,3cdKpτKq×dKqτKp+∑pq∈E3,5cdKpτKq×dKqτKp+∑pq∈E4,5cdKpτKq×dKqτKp+∑pq∈E4,4cdKpτKq×dKqτKp+∑pq∈E3,4cdKpτKq×dKqτKp+∑pq∈E4,4c∗dKpτKq×dKqτKp+∑pq∈E4,6cdKpτKq×dKqτKp+∑pq∈E6,6cdKpτKq×dKqτKp+∑pq∈E5,8cdKpτKq×dKqτKp=E2,3c2×3×2×2+E3,3c2×3×2×3+E3,5c2×5×3×3+E4,5c2×5×3×4+E4,4c2×4×3×4+E3,4c2×4×3×3+E4,4c∗3×4×3×4+E4,6c3×6×3×4+E6,6c3×6×3×6+E5,8c3×8×4×5=4n+1224+6n+236+4n+1290+12n+4120+36n+1296+12n+472+6n+2144+12n+4216+3n+1324+8n+8480=96n+288+216n+72+360n+1080+1440n+480+3456n+1152+864n+288+864n+288+2592n+864+972n+324+3840n+3840=14700n+8676.


## 5. Comparisons and Conclusions

In this section, we compare zinc oxide (H) and zinc silicate (K) related MONs via some Zagreb connection indices (ZCIs) such as first ZCI, second ZCI, modified first ZCI, modified second ZCI, modified third ZCI, and modified fourth ZCI with the help of Tables9–14 that have been constructed by using numerical values of the aforementioned ZCIs. The graphical presentations for ZCIs of MONs are presented in Figures 3–10.Table 9
Numerical values ofZC1 for H and K networks on dimensions 3≤n≤10.
ZCIs345678910ZC1H3896496460327100816892361030411372ZC1K521666808144960811072125361400015464Table 10
Numerical values ofZC2 for H and K networks on dimensions 3≤n≤10.
ZCIs345678910ZC2H502464007776915210528119041328014656ZC2K68048714106241253414444163541826420174Table 11
Numerical values ofZC1∗ for H and K networks on dimensions 3≤n≤10.
ZCIs345678910ZC1∗H24483120379244645136580864807152ZC1∗K31284004488057566632750883849260Table 12
Numerical values ofZC2∗ for H and K networks on dimensions 3≤n≤10.
ZCIs345678910ZC2∗H6344808498241156413304150441678418524ZC2∗K834410684130241536417704200442238424724Table 13
Numerical values ofZC3∗ for H and K networks on dimensions 3≤n≤10.
ZCIs345678910ZC3∗H66088416102241203213840156481745619264ZC3∗K856810964133601575618152205482294425340Table 14
Numerical values ofZC4∗ for H and K networks on dimensions 3≤n≤10.
ZCIs345678910ZC4∗H382564860058944692887963289976100320110664ZC4∗K52776674768217696876111576126276140976155676Figure 3
Comparison ofZC1 index for H and K networks on dimensions 3≤n≤10.Figure 4
Comparison ofZC2 index for H and K networks on dimensions 3≤n≤10.Figure 5
Comparison ofZC1∗ index for H and K networks on dimensions 3≤n≤10.Figure 6
Comparison ofZC2∗ index for H and K networks on dimensions 3≤n≤10.Figure 7
Comparison ofZC3∗ index for H and K networks on dimensions 3≤n≤10.Figure 8
Comparison ofZC4∗ index for H and K networks on dimensions 3≤n≤10.Figure 9
Comparison of ZCIs of networkH on dimensions 3≤n≤10.Figure 10
Comparison of ZCIs of networkK on dimensions 3≤n≤10.The comparative study of zinc-related MONs is highlighted by the following conclusions:(i)
From Tables9–14 and Figures 3–8, we see that the behaviors for all the ZCIs of zinc silicate MONs have more values and upper lines than zinc oxide MONs with the following order:ZC1K≥ZC1H, ZC2K≥ZC2H, ZC1∗K≥ZC1∗H, ZC2∗K≥ZC2∗H, ZC3∗K≥ZC3∗H, and ZC4∗K≥ZC4∗H.(ii)
From Tables15 and 16 and Figures 9 and 10, we see that modified fourth ZCI ZC4∗ attains more values and upper lines than other ZCIs for both zinc-related MONs.(iii)
The modified first Zagreb connection indexZC1∗ attained better values of correlation coefficient for the thirteen physicochemical properties of octane isomers than other classical Zagreb indices. In this paper, novel connection-based Zagreb index ZC4∗ attains better values of correlation coefficient for increasing order in both the cases of zinc-related MONs.(iv)
Table17 shows that zinc silicate-related MON of dimension n for the aforesaid ZCIs has attained upward position than zinc oxide-related MON.(v)
Moreover, these general relations (Tables15–17) indicate that the chemical capability of zinc silicate-related MON is better than zinc oxide-related MON for all values of n.Now, the problem is still open for prism, product, subdivision, and their compliment networks with the help of connection-based Zagreb indices.Table 15
Numerical table for indicated ZCIs of networkH on dimensions 3≤n≤10.
ZCIsZC1HZC2HZC1∗HZC2∗HZC3∗HZC4∗H338965024244863446608382564496464003120808484164860056032777637929824102245894467100915244641156412032692887816810528513613304138407963289236119045808150441564889976910304132806480167841745610032010113721465671521852419264110664Table 16
Numerical table for indicated ZCIs of networkK on dimensions 3≤n≤10.
ZCIsZC1KZC2KZC1∗KZC2∗KZC3∗KZC4∗K352166804312883448568527764668087144004106841096467476581441062448801302413360821766960812534575615364157569687671107214444663217704181521115768125361635475082004420548126276914000182648384223842294414097610154642017492602472425340155676Table 17
Comparison of indicated ZCIs for alln.
ZCIsK − H = ZNSL (n) − ZNOX (n)ResultsFirst ZCI396n + 132K > HSecond ZCI534n + 178K > HModified first ZCI204n + 68K > HModified second ZCI600n + 200K > HModified third ZCI588n + 196K > HModified fourth ZCI4356n + 1452K > H



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*Source: 5066394-2021-10-18.xml*