# Studies of Metal Organic Networks via M-Polynomial-Based Topological Indices

Authors: Muhammad Tanveer Hussain; Muhammad Javaid; Sajida Parveen; Hafiz Muhammad Awais; Md Nur Alam
Journal: Journal of Mathematics
 (2022)

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## Abstract

Topological index (TI) is a graph-theoretic tool that is used to study different physical and structural properties of the networks in various disciplines of science such as computer science, chemistry, and information technology. In this article, we study transition metal tetra-cyano-benzene organic networks by computing their M-polynomials and various topological indices (TIs). At the end, a comparison is also included between all the computed degree-based topological indices to show their betterness.


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## Body

## 1. Introduction

Metal-organic networks (MONs) is a modern class of porous materials constructed with organic linkers based on secondary building units and metal clusters, with significant attention in the last years. Consequently, their structural diversity, high-grade porosity, and compositional tenability represent the exciting of the various physicochemical characteristics such as the grafting active group [1], postsynthetic ligand, changing organic ligand [2], preparing composites with various substances [3, 4], ions exchange [5], and impregnating suitable active materials [6], methane, and hydrogen storage materials that involve magnesium-decorated fullerenes within metal-organic frameworks [7, 8]. Seetharaj et al. [9] studied the relationship among different solvents, architecture, temperature, molar ratio, and pH of the MONs. Therefore, MONs are also studied for the purification and separation of various liquids and gases, energy storage tools in batteries [10], and precursor for the formation of nanostructures [11–13] and later on, the idea of composition of linkers with the assistance of MONs for topological insight [14–16].Graph theory presents the modern tools which are used to study the chemical compounds and to discuss their properties. The calculated TIs of the molecular graphs are the numeric values that describe chemical reactivity, biological activity, and physical property of chemical compounds such as heat of evaporation and formation; melting, boiling and flash points, surface tension, retention time in chromatography, pressure, temperature, density, and partition coefficients [17]. Bruckler et al. [18] considered Q-indices that consist of contributions of vertex pairs that exponentially decrease with distance; Gonzalez-Diaz et al. [19] reviewed and commented on the “quo vadis” and challenges in the definition of TIs as we enter the new century; Klavzar and Gutman [20] used the bound 4W<MTI<6.93W that implies that W and MTI are linearly correlated not only in the case of acyclic molecules but also in the case of molecules with arbitrarily many cycles; Matamala and Estrada [21] provided the role of topological structure in the study of molecular physicochemical properties; Randic [22] reported that the sensitivity of TIs for structural changes within comprehensive groups of cyclic saturated hydrocarbons is evaluated. For more details, see [23–25].Moreover, Wiener defined a distance-based topological index in graph theory for the boiling point of paraffin [26]. Gutman and Trinajstic [27] defined TIs named as 1st and 2nd Zagreb indices to compute the total π-electrons energy of different organic molecules [27]. TIs represent a valuable role in the quantitative structure-activity/property relationships to characterize a graph (network) with a biological and chemical property or activity. This mathematical relationship is represented as P=fM, where P is an activity or property and M is any network [28, 29]. Therefore, TIs of many networks such as silicate, honeycomb, and hexagonal networks [30], rhombous silicate and rhombus oxide networks [31], titania nanotube [32], superconducting materials [33], MOFs [34], hexagonal parallelogram nanotube [35], fullerene networks and carbon nanotube [36], regular hexagonal lattice [37], graphs with given number of cut vertices [38], and explore trees in terms of given number of vertices of maximum degree [39] are studied in the literature.In this article, we solve the new M-polynomials, and by using these M-polynomials, the different TIs are calculated for the metal-organic networks. In the end, some comparisons between the computed TIs with the help of graphs are also included. The remaining portion is arranged as follows: Section2 contains various definitions as well as techniques that are used in calculated results. Sections 3 and 4 present the main results of M-polynomials which are utilized in topological indices. Section 5 involves the numerical and graphical demonstration and final remarks.


## 2. Preliminaries

A networkG=VG,EG, where VG=x1,x2,x3,…,xn and EG⊆VG×VG are the organic ligands (metal node) and bonds between the nodes of G, respectively. VG=v and EG=e are defined as the order and size of a graph G. In connected and simple graphs, a path is occurring between two atoms (vertices) and the distance between two atoms x and y is the length of the shortest path that is shown as φx,y, in G [22, 40].1st and 2nd Zagreb indices: letG be a simple graph; then, its 1st and 2nd Zagreb indices are(1)M1G=∑x∈VGφx2=∑xy∈EGφx+φy,M2G=∑xy∈EGφx×φy.General Randic index: for a graphG, set of real number R, and β∈R, the general Randi c′ index is(2)RβG=∑xy∈EGφxφyβ.Symmetric division deg index: letG be a simple and connected graph; then, the symmetric division deg index is(3)SDDG=∑xy∈EGminφx,φymaxφx,φy+maxφx,φyminφx,φy.Harmonic index: letG be a simple graph; then, its harmonic index is(4)HG=∑xy∈EG2φx+φy.Inverse sum index: letG be a simple graph; the inverse sum index is(5)ISG=∑xy∈EGφxφyφx+φy.Augmented Zagreb index: letG be a simple graph; then, the augmented Zagreb index is(6)AZIG=∑xy∈EGφx×φyφx+φy−23.M-polynomial: for a graphG and ms,tG,s,t≥1 be the number of bonds (edges) e=xy of G in such a way φxφy=g,h. Then, M-polynomial of G is(7)MG,g,h=∑s≤tms,tGgsht.In Tables1 and 2, the relationship among the aforesaid TIs and the M-polynomial is given.Table 1
Derivation of TIs from M-polynomial.
Indicesfg,hDerivation fromMG,g,hM1g+hDg+DhMG,g,h|g=1=hM2ghDgDhMG,g,h|g=1=hMM21/ghSgShMG,g,h|g=1=hRβghβ,β∈NDgβDhβMG,g,h|g=1=hRβRβ1/ghβ,β∈NSgβShβMG,g,h|g=1=hSDDg2+b2/ghDgSh+DhSgMG,g,h|g=1=hTable 2
Some other TIs from M-polynomial.
Indicesfg,hDerivation fromMG,g,hH2/g+h2SgJMG,g,h|g=1=hISgh/g+hSgQ2JDgDhMG,g,h|g=1=hAZIgh/g+h−23Sg3JDg3Dh3MG,g,h|g=1=hMoreover,Dg=ϕfg,h/ϕg, Dh=ϕfg,h/ϕh, Sg=∫0gϕft,h/tdt, Sh=∫0hϕfg,t/tdt, Jfg,h=fg,g, and Qβfg,h=gβfg,h, where β≠0. For further detailed discussion, refer to [41]. Metal-organic networks are considered such types of networks that are investigated for constant porosity. Therefore, these porous materials are filled with gas or liquid. Such types of pores are made with the help of metal nodes and organic ligands. The organic molecules are organic ligands that might be held together to form coordination. In organic chemistry, ligands are recognized as molecules or ions that are related to different types of functional groups attached to a central atom to form complicated coordination, i.e., coordinate covalent bonds between the central atom and ligands. These networks with porosity form in such a way that ligands attached in different places and metals place only in the middle of the network. Moreover, MONs are considered to be in two or three-dimensional networks. In the current discussion, we are dealing with MON, TM-TCNB (TM, transition metals of the 3rd series: titanium, vanadium, chromium, and zinc, where transition metals (group B elements) occupy the middle section (d-block) of the periodic table; TCNB, tetra-cyano-benzene). The two-dimensional MON of TM-TCNB is shown in Figure 1 which represents four colors consisting of blue, cyano, red, and yellow, representing N, TM, H, and N, respectively. Therefore, N stands for nitrogen, TM for transition metals, H for hydrogen, and C for carbon atom. The degree-based TI is a modern tool that opens up exciting discussion for many networks in the field of degree-based TIs. The modern contribution of these TIs based on degree is to compute the exact form of more than ten degrees and connection number-based descriptors.Figure 1
Transition metal tetra-cyano-benzene organic network.Now, we take TM-TCNB as the primary network, and it does not recognize double or triple bonds (that is the primary reason to eliminate the hydrogen atoms). Furthermore, we eliminated eight hydrogen atoms from the network of TM-TCNB that were available at paranodes (vertices) of every benzene ring (present at the middle of every metal). After eliminating hydrogen’s atoms from the middle of the benzene ring, the rest of the network is presented as (TM-TCNB)1, and this is our first metal-organic network. After that, we are going through the importance of the gained network of (TM-TCNB)1. So, we are left at the point to add four new benzene rings (adjacent to middle benzene into first network (TM-TCNB)1). Now, the resultant structure is TM-TCNB2, and we know it as our second metal-organic network, as shown in Figure 2.Figure 2
Transition metal tetra-cyano-benzene organic network type-I.


## 3. Main Results

In this section, we find theM-polynomial and different TIs with the help of this M-polynomial for the of the transition metal tetra-cyano-benzene organic network type-I (TM-TCNB)1 and type-II (TM-TCNB)2.Theorem 1.
LetG1=TM−TCNB1 be a transition metal tetra-cyano-benzene organic network type-I. Then, the M-polynomial of G1 is(8)MG1,g,h=8z+8g2h2+56z−8g2h3+60z−20g3h3+16zg3h4.Proof.
From Figure2, we note that 3 distinct types of vertices exist in (TM-TCNB)1 as follows:(9)V1=x∈VG1|φx=2,V2=x∈VG1|φx=3,V3=x∈VG1|φx=4,such that(10)V1=36z+4,V2=64z−16,V3=4z.
Moreover,(11)VTM−TCNB1n=V1+V2+V3=104z−12.
We have 4 distinct types of edges that is based on the degrees of end vertices inG1 that are 2,2,2,3,3,3,3,4, such that(12)E2,2=e=xy∈EG1|φx=2,φy=2,E2,3=e=xy∈EG1|φx=2,φy=3,E3,3=e=xy∈EG1|φx=3,φy=3,E3,4=e=xy∈EG1|φx=3,φy=4withE1=8z+8,E2=56z−8,E3=60z−20,E4=16z.Moreover,EG1=E1+E2+E3+E4=140z−20.
Now, the partition of the vertex set and edge set of the type-1 MON are given in Tables3 and 4.
Now, by using definitions of M-polynomial Tables3 and 4, we have(13)MG1,g,h=∑s≤tEs,tG1gsht=∑2≤2E2,2G1g2h2+∑2≤3E2,3G1g2h3+∑3≤3E3,3G1g3h3+∑3≤4E3,4G1g3h4=E1g2h2+E2g2h3+E3g3h3+E4g3h4=8z+8g2h2+56z−8g2h3+60z−20g3h3+16zg3h4.Table 3
The partition of the vertex set ofTM−TCNB1.
Vertex partitionV1V2V3Cardinality36z+464z−164zTable 4
The partition of the edge set ofTM−TCNB1.
Edge partitionE1E2E3E4Cardinality8z+856z−860z−2016zTheorem 2.
LetG1=TM−TCNB1 be the transition metal tetra-cyano-benzene organic network type-I. Then,(14)M1G1=784z−128,M2G1=1100z−196,MM2G1=20z−149,RβG1=4β8z+8+16β56z−8+9β60z−20+12β16z,RRβG1=14β8z+8+16β56z−8+19β60z−20+112β16z,SDDG1=8723z−1243.Proof.
Letfg,h=MG1,g,h be the M-polynomial of the type-1 (Theorem 1) MON; then,(15)fg,h=8z+8g2h2+56z−8g2h3+60z−20g3h3+16zg3h4.
First, we find the required partial derivatives and integrals as(16)Dgfg,h=∂fg,h∂g=28z+8gh2+256z−8gh3+360z−20g2h3+316zg2h4,Dhfg,h=∂fg,h∂h=28z+8g2h+356z−8g2h2+360z−20g3h2+416zg3h3,DgDhfg,h=48z+8gh+656z−8gh2+960z−20g2h2+1216zg2h3,Tgfg,h=∫0gfu,hudufu,h=8z+8u2h2+56z−8u2h3+60z−20u3h3+16zu3h4fu,hu=8z+8uh2+56z−8uh3+60z−20u2h3+16zu2h4∫0gfu,hudu=8z+82u2h2+1256z−8u2h3+1360z−20u3h3+1316zu3h40g=128z+8g2h2+1256z−8g2h3+1360z−20g3h3+1316zg3h4,Thfg,h=∫0hfg,uudufg,u=8z+8g2u2+56z−8g2u3+60z−20g3u3+16zg3u4fg,uu=8z+8g2u+56z−8g2u2+60z−20g3u2+16zg3u3∫0hfg,uudu=128z+8g2u2+1356z−8g2u3+1360z−20g3u3+1416zg3u40h=128z+8g2h2+1356z−8g2h3+1360z−20g3h3+1416zg3h4,TgThfg,h=TgThfg,h=128z+8u2h2+1356z−8u2h3+1360z−20u3h3+1416zu3h4=128z+8uh2+1356z−8uh3+1360z−20u2h3+1416zu2h4=148z+8u2h2+1656z−8u2h3+1960z−20u3h3+1816zu3h40g=148z+8g2h2+1656z−8g2h3+1960z−20g3h3+1816zg3h4,DhTgfg,h=Dh128z+8g2h2+1256z−8g2h3+1360z−20g3h3+1316zg3h4=228z+8g2h+3256z−8g2h2+3360z−20g3h2+4316zg3h3=8z+8g2h+3256z−8g2h2+60z−20g3h2+4316zg3h3,DgThfg,h=Dg128z+8g2h2+1356z−8g2h3+1360z−20g3h3+1416zg3h4=8z+8gh2+2356z−8gh3+60z−20g2h3+3416zg3h3.DgβDhβfg,h=4β8z+8+6β56z−8gh2+9β60z−20g2h2+12β16zg2h3TgβThβfg,h=8z+84βg2h2+16β56z−8g2h3+19β60z−20g3h3,+112β16zg3h4.
Now, by substitutingh=g=1,(17)Dgfg,hg=1=h=356z−60,Dhfg,hg=1=h=428z−68,DgDhfg,hg=1=h=1100z−196,Tgfg,hg=1=h=1723z−203,Thfg,hg=1=h=1403z−163,TgThfg,hg=1=h=20z−149,DhTgfg,hg=1=h=5203z−24,DgThfg,hg=1=h=3523z−523,DgβDhβfg,hg=1=h=4β8z+8+6β56z−8+9β60z−20+12β16z,TgβThβfg,hg=1=h=8z+84β+16β56z−8+19β60z−20+112β16z.
Using these values in formulae of Table1,(18)M1G1=Dg+Dhfg,hg=1=h=Dgfg,hg=1=h+Dhfg,hg=1=h=784z−128,M2G1=DgDhfg,hg=1=h=DgDhfg,hg=1=h=1100z−196,MM2G1=TgThfg,hg=1=h=20z−149,RβG1=DgβDhβfg,hg=1=h=4β8z+8+6β56z−8+9β60z−20+12β16z,RRβG1=TgβThβfg,hg=1=h=8z+84β+16β56z−8+19β60z−20+112β16z,SDDG1=DgTh+DhTgfg,hg=1=h=DgThfg,hg=1=h+DhTgfg,hg=1=h=8723z−1243.Theorem 3.
LetG1=TM−TCNB1 be the transition metal tetra-cyano-benzene organic network type-I. Then,(19)HG1=178435z−8815,ISG1=674235z−1585,AZIG1=1037635z−3965.Proof.
Letfg,h=MG1,g,h be the M-polynomial of MON; then,(20)MG1,g,h=8z+8g2h2+56z−8g2h3+60z−20g3h3+16zg3h4Jfg,h=8z+8g4+56z−8g5+60z−20g6+16zg7SgJfg,h=148z+8g4+1556z−8g5+1660z−20g6+1716zg7JDgDhfg,h=48z+8g2+656z−8g3+960z−20g4+1216zg8Q2JDgDhfg,h=48z+8g4+656z−8g5+960z−20g6+1216zg7SgQ2JDgDhfg,h=8z+8g4+6556z−8g5+9660z−20g6+12716zg7JDg3Dh3fg,h=438z+8g2+6356z−8g3+9360z−20g4+12316zg5Sg3JDg3Dh3fg,h=4238z+8g2+63356z−8g3+94360z−20g4+125316zg5SgJfg,hg=1=h=89235z−4415SgQ2JDgDhfg,hg=1=h=674235z−1585Sg3JDg3Dh3fg,hg=1=h=1037635z=3965Jfg,h=Jfg,h=8z+8g4+56z−8g5+60z−20g6+16zg5SgJfg,h=∫0gfJfg,hudu=∫0g8z+8u4+56z−8u5+60z−20u6+16zu7udu=∫0g8z+8u3+56z−8u4+60z−20u5+16zu7du=8z+8u44+1556z−8u5+1660z−20u6+16z7u70g=8z+8g44+1556z−8g5+1660z−20g6+16z7g7JDgDhfg,h=48z+8g2+656z−8g3+960z−20g4+1216zg5Q2JDgDhfg,h=Q248z+8g2+656z−8g3+960z−20g4+1216zg5=48z+8g4+656z−8g5+960z−20g6+1216zg7SgQ2JDgDhfg,h=Sg48z+8g4+656z−8g5+960z−20g6+1216zg7=∫0g48z+8u3+656z−8u4+960z−20u5+1216zu6du=448z+8u4+6556z−8u5+9660z−20u6+12716zu70g=8z+8g4+6556z−8g5+3260z−20g6+12716zg7JDg3Dh3fg,h=438z+8g2+6356z−8g3+9360z−20g4+12316zg5Sg3JDg3Dh3fg,h=438z+8u2+6356z−8u3+9360z−20u4+12316zu5=∫0g438z+8u+6356z−8u2+9360z−20u3+12316zu4du=6428z+8u2+216356z−8u3+729460z−20u4+1728516zu50g=6428z+8g2+216356z−8g3+729460z−20g4+1728516zg5,therefore,(21)SgJfg,hg=1=h=148z+8+1556z−8+1660z−20+167z=89235z−4415,SgQ2JDgDhfg,hg=1=h=8z+8+6556z−8+3260z−20+12716z=674235z−1585,Sg3JDg3Dh3fg,hg=1=h=6428z+8+216356z−8+729460z−20+1728516z=1037635z−3965,HG1=2SgJfg,hg=1=h=289235z−4415=178435z−8815,ISG1=SgQ2JDgDhfg,hg=1=h=674235z−1585,AZIG1=Sg3JDg3Dh3fg,hg=1=h=1037635z−3965.Theorem 4.
LetG2=TM−TCNB2 be the transition metal tetra-cyano-benzene organic network type-II. Then, the M-polynomial of G2 is(22)MG2,g,h=8z+1g2h2+165z−1g2h3+116z−2g3h3+16zg3h4.Proof.
From Figure3, we show the partition of the vertex set and the edge set of G2 according to degrees of the vertices. In this way, we get 3 distinct types of vertices in G2 as follows:(23)V1=x∈VG2|φx=2,V2=x∈VG2|φx=3,V3=x∈VG2|φx=4,such that(24)V1=48z,V2=56z−20,V3=4z.
Also,(25)G2=x=V1+V2+V3=128z−20.
Now, we get distinct types of edges based on the degree of end vertices ofG2 that are(26)2,2,2,3,3,3&3,4E2,2=e=xy∈EG2|φx=2,φy=2,E2,3=e=xy∈EG2|φx=2,φy=3,E3,3=e=xy∈EG2|φx=3,φy=3,E3,4=e=xy∈EG2|φx=3,φy=4withEG2=e=E2,2+E2,3+E3,3+E3,4=170z−30.
Therefore, partition of the vertex set and the edge set of the metal-organic networkTM−TCNB2 are given in Tables 5 and 6.
By using definition of M-polynomial, Tables5 and 6, we have(27)MG2,g,h=∑s≤tEs,tG2gsht=∑2≤2E2,2G2g2h2+∑2≤3E2,3G2g2h3+∑3≤3E3,3G2g3h3+∑3≤4E3,4G2g3h4=E1g2h2+E2g2h3+E3g3h3+E4g3h4.MG2,g,h=8z+1g2h2+165z−1g2h3+116z−2g3h3+16zg3h4.Table 5
The partition of the vertex set ofTM−TCNB2.
Vertex partitionV1V2V3Cardinality48z76z−204zTable 6
The partition of the edge set ofTM−TCNB2.
Edge partitionE2,2E2,3E3,3E3,4Cardinality8z+1165z−1116z−216zFigure 3
Transition metal tetra-cyano-benzene organic network type-II.Theorem 5.
LetG2=TM−TCNB2 be a transition metal tetra-cyano-benzene organic network type-II. Then,(28)M1G2=940z−180,M2G2=1298z−98,MM2G2=24z−289,RβG2=4β8z+1+6β165z−1+9β116z−2+12β16z,RRβG2=84βz+1+166β5z−1+119β6z−2+112β16z.Proof.
Letfg,h=MG2,g,h be the M-polynomial of second MON:(29)MG2,g,h=8z+1g2h2+165z−1g2h3+116z−2g3h3+16zg3h4.
The needed derivatives and integrals are calculated as(30)Dgfg,h=∂fg,h∂g=16z+1gh2+325z−1gh3+336z−2g2h3+316zg2h4Dgfg,hg=1=h=16z+1+325z−1+336z−2+316z=422z−82,Dhfg,h=∂fg,h∂h=16z+1g2h+485z−1g2h2+336z−2g3h2+416zg2h3Dhfg,hg=1=h=16z+1+485z−1+336z−2+416z=518z−98,DgDhfg,h=32z+1gh+965z−1gh2+996z−2g2h2+1216zg2h3DgDhfg,hg=1=h=32z+1+965z−1+996z−2+1216z=1298z−262,Sgfg,h=∫0gfu,hudu=∫0g8z+1u2h2+165z−1u2h3+116z−2u3h3+16zu3h4udu∫0gfu,hudu=8z+12u2h2+1625z−1u2h3+1136z−2u3h3+1316zu3h40g=82z+1g2h2+1625z−1g2h3+1136z−2g3h3+1316zg3h4,Sgfg,hg=1=h=82z+1+1625z−1+1136z−2+1316z=2143z−343,Shfg,h=∫0hfg,uudufg,u=8z+1g2u2+165z−1g2u3+116z−2g3u3+16zg3u4fg,uu=8z+1g2u+165z−1g2u2+116z−2g3u2+16zg3u3∫0hfg,uudu=82z+1g2u2+1635z−1g2u3+1136z−2g3u3+1416zg3u40h=82z+1g2h2+1635z−1g2h3+1136z−2g3h3+1416zg3h4Shfg,hg=1=h=82z+1+1635z−1+1136z−2+1416z=1703z−263,SgShfg,h=SgShfg,h=2z+1g2h2+835z−1g2h3+1196z−2g3h3+11216zg3h4SgShfg,hg=1=h=2z+1+835z−1+1196z−2+11216z=24z−289,DhSgfg,h=Dh82z+1g2h2+1625z−1g2h3+1136z−2g3h3+1316zg3h4=282z+1g2h+31625z−1g2h2+31136z−2g3h2+4316zg3h3=8z+1g2h+245z−1g2h2+116z−2g3h2+4316zg3h3DhSgfg,hg=1=h=8z+1+245z−1+116z−2+4316z=6463z−38,DgShfg,h=Dg82z+1g2h2+1635z−1g2h3+1136z−2g3h3+1416zg3h4=8z+1gh2+3235z−1gh3+116z−2g2h3+3416zg2h4DgShfg,hg=1=h=8z+1+3235z−1+116z−2+3416z=4183z−743,DgβDhβfg,h=4β8z+1gh+6β165z−1gh2+9β116z−2g2h2+12β16zg2h3DgβDhβfg,hg=1=h=4β8z++6β165z−1+9β116z−2+12β16zSgβShβfg,h=8z+14βg2h2+166β5z−1g2h3+119β6z−2g3h3+112β16zg3h4SgβShβfg,hg=1=h=8z+14β+166β5z−1+119β6z−2+112β16z.
Now, by substitutingg=h=1,(31)Dgfg,hg=1=h=422z−82,Dhfg,hg=1=h=518z−98,DgDhfg,hg=1=h=1298z−262,Sgfg,hg=1=h=2143z−343,Shfg,hg=1=h=1703z−263,SgShfg,hg=1=h=24z−289,DhSgfg,hg=1=h=6463z−38,DgShfg,hg=1=h=4183z−743,DgβDhβfg,hg=1=h=4β8z+1+6β165z−1+9β116z−2+12β16z,SgβShβfg,hg=1=h=8z+14β+166β5z−1+119β6z−2+112β16z,M1G2=Dg+Dhfg,hg=1=h=Dgfg,hg=1=h+Dhfg,hg=1=h=940z−180,M2G2=DgDhfg,hg=1=h=DgDhfg,hg=1=h=1298z−98,MM2G2=SgShfg,hg=1=h=24z−289,RβG2=DgβDhβfg,hg=1=h=4β8z+1+6β165z−1+9β116z−2+12β16z,RRβG2=SgβShβfg,hg=1=h=8z+14β+166β5z−1+119β6z−2+112β16z,SDDG2=DgSh+DhSgfg,hg=1=h=DgShfg,hg=1=h+DhSgfg,hg=1=h=10643z−1883.Theorem 6.
LetG2=TM−TCNB2 be a transition metal tetra-cyano-benzene organic network type-II. Then,(32)HG2=4387z−14615,ISG2=16137z−2215,AZIG2=67078614000z−1006732.Proof.
Letfg,h=MG2,g,h be the M-polynomial of second MON; then,(33)MG2,g,h=8z+1g2h2+165z−1g2h3+116z−2g3h3+16zg3h4,Jfg,h=8z+1g4+165z−1g5+116z−2g6+16zg7,SgJfg,h=84z+1g4+1655z−1g5+1166z−2g6+1716zg7,JDgDhfg,h=32z+1g2+965z−1g3+996z−2g4+1216zg5,Q2JDgDhfg,h=32z+1g4+965z−1g5+996z−2g6+1216zg7,SgQ2JDgDhfg,h=324z+1g4+96755z−1g5+9966z−2g6+12716zg7Dg3Dh3fg,h=438z+1gh+163165z−1gh2+93116z−2g2h2+12316zg2h3,JDg3Dh3fg,h=438z+1g2+163165z−1g3+93116z−2g4+12316zg5,Sg3JDg3Dh3fg,h=4238z+1g2+633165z−1g3+943116z−2g4+125316zg5,SgJfg,hg=1=h=2197z−7315,SgQ2JDgDhfg,hg=1=h=16137z−2215,Sg3JDg3Dh3fg,hg=1=h=67078614000z−1006732,Jfg,h=8z+1g4+165z−1g5+116z−2g6+16zg7,SgJfg,h=∫0gfJfg,hudu=∫0g8z+1u4+165z−1u5+116z−2u6+16zu7udu=∫0g8z+1u3+165z−1u4+116z−2u5+16zu6du=8z+1u44+1655z−1u5+1166z−2u6+16z7u70g=8z+1g44+1655z−1g5+1166z−2g6+16z7g7,JDgDhfg,h=32z+1g2+965z−1g3+996z−2g4+1216zg5,Q2JDgDhfg,h=Q232z+1g2+965z−1g3+996z−2g4+1216zg5=32z+1g4+965z−1g5+996z−2g6+1216zg7,SgQ2JDgDhfg,h=Sg32z+1g4+965z−1g5+996z−2g6+1216zg7=∫0g32z+1g4+965z−1g5+996z−2g6+1216zg7du=324z+1u4+1655z−1u5+1166z−2u6+12716zu70g=8z+1g4+9655z−1g5+9966z−2g6+12716zg7,JDg3Dh3fg,h=438z+1g2+163165z−1g3+93116z−2g4+12316zg5,Sg3JDg3Dh3fg,h=Sg3438z+1g2+163165z−1g3+93116z−2g4+12316zg5=∫0g438z+1g2+163165z−1g3+93116z−2g4+12316zg5du=8423z+1g2+166335z−1g3+943116z−2g4+125316zg5.
We get(34)SgJfg,hg=1=h=8z+1g44+1655z−1g5+1166z−2g6+16z7g7=8z+14+1655z−1+1166z−2+16z7=2197z−7315,SgQ2JDgDhfg,hg=1=h=8z+1g4+9655z−1g5+9966z−2g6+12716zg7=8z+1+9655z−1+9966z−2+12716z=16137z−2215,Sg3JDg3Dh3fg,hg=1=h=8423z+1g2+166335z−1g3+943116z−2g4+125316zg5=8423z+1+166335z−1+943116z−2+125316z=67078614000z−1006732.
Now, using Table2,(35)HG2=2SgJfg,hg=1=h=22197z−7315=4387z−14615,ISG2=SgQ2JDgDhfg,hg=1=h=16137z−2215,AZIG2=Sg3JDg3Dh3fg,hg=1=h=67078614000z−1006732.


## 4. Comparison and Conclusion

In this section, we calculate various degree-based topological indices and present the results in the form of tables and graphs as given Tables7–10.Table 7
Comparison among different TIs of (TM-TCNB)1.
zM1G1M2G1MM1G1SDDG1165890417.7778249.333321440200437.111154032224310456.4444830.666743008420475.77781121.333353792530495.11111412645766404114.44441702.6667753607504133.77781993.3333861448604153.11112284969289704172.44442574.666710771210804191.77782865.3333Table 8
Comparison between different TIs of (TM-TCNB)2.
ZM1G2M2G2MM1G2SDDG21760120020.888929221700249844.8889646.666732640379668.88891001.333343580509492.88891356.0000545206392116.88891710.6667654607690140.88892065.3333764008988164.88892420.00008734010286188.88892774.66679828011584212.88893129.333310922012882236.88893484Table 9
Comparison among different TIs of (TM-TCNB)1.
zHG1ISG1AZIG1145.1048161.028616787.6296.0762353.657137540.23147.0476546.285758292.84198.0190738.914379045.45248.9905931.5429997986299.96191124.1714120550.67350.93331316.8141303.28401.90481509.4286162055.89452.87621702.0571182808.410503.84761894.6857203561Table 10
Comparison among different indices of (TM-TCNB)2.
zHG2ISG2AZIG2152.8381186.22861362.37152115.4095416.65713039.33673177.9810647.08574716.30204240.5524877.51436393.26725303.12381107.94298070.23256365.69521338.3719747.19777428.26671568.800011424.16308490.83811799.228613101.12829553.40952029.657114778.093510715.98102260.085716455.0587


### 4.1. Comparison betweenM1G1M2G1, MM1G1, and SDDG1 of (TM-TCNB)1

The comparison of first ZagrebM1G1, second Zagreb M2G1, second modified Zagreb MM1G1, and symmetry division deg SDDG1 of (TM-TCNB)1 is computationally computed with the help of M-polynomials, as given in Table 3. We also computed these indices for different values of z. When z increases, then all TIs are increased in the same order. We show the graphical presentation of all the aforesaid topological indices in Figure 4 for the particular values of z.Figure 4
Comparison of computed TIs of (TM-TCNB)1.


### 4.2. Comparison betweenM1G2M2G2, MM1G2, and SDDG2 of (TM-TCNB)2

The comparison of first ZagrebM1G1, second Zagreb M2G1, second modified Zagreb MM1G1, and symmetry division deg SDDG1 of (TM-TCNB)2 is computationally computed with the assistance of aforesaid M-polynomials given in Table 4. We calculated these indices for different values of z. When z increases, then all the TIs are increased in the same order. We show the graphical representation of all the degree-based TIs in Figure 5 for the particular values of z.Figure 5
Comparison between different Zagreb indices of (TM-TCNB)2.


### 4.3. Comparison betweenHG1, ISG1, and AZIG1 of (TM-TCNB)1

The comparison of the harmonicH, inverse sum (IS), and augmented Zagreb index (AZI) of (TM-TCNB)1 is computationally computed with the assistance of M-polynomial given in Table 5. We calculated these indices for different values of z. When z increases, then all the TIs are increased in the same order. We show the graphical representation of all the degree-based TIs in Figure 6 for the particular values of z.Figure 6
Comparison among different TIs of (TM-TCNB)1.


### 4.4. Comparison betweenHG2, ISG2, and AZIG2 of (TM-TCNB)2

The comparison of the harmonicHG2, inverse sum ISG2, and augmented Zagreb index AZIG2 of (TM-TCNB)2 is computationally computed with the assistance of M-polynomial given in Table 6. We calculated these indices for different values of z. When z increases, then all the TIs are increased in the same order. We show the graphical representation of all the degree-based TIs in Figure 7 for the particular values of z.Figure 7
Comparison among different indices of (TM-TCNB)2.In this work, we discussed the M-polynomials of (TM-TCNB)1 and (TM-TCNB)2. We also work out on the modern degree-based TIs, including 1st Zagreb M1, 2nd Zagreb M2, 2nd modified Zagreb MM2, general Randic RRγ, symmetry division deg (SDD), harmonic H, inverse sum (IS), and the augmented Zagreb index (AZI) and showed their behavior for (TM-TCNB)1 and (TM-TCNB)2. In addition, all the obtained results are also compared with the help of the numerical values provided in the tables, (see Tables 7–10). The problem is still open for the computational results related to different topological indices of the various metal-organic networks.


## 4.1. Comparison betweenM1G1M2G1, MM1G1, and SDDG1 of (TM-TCNB)1

The comparison of first ZagrebM1G1, second Zagreb M2G1, second modified Zagreb MM1G1, and symmetry division deg SDDG1 of (TM-TCNB)1 is computationally computed with the help of M-polynomials, as given in Table 3. We also computed these indices for different values of z. When z increases, then all TIs are increased in the same order. We show the graphical presentation of all the aforesaid topological indices in Figure 4 for the particular values of z.Figure 4
Comparison of computed TIs of (TM-TCNB)1.


## 4.2. Comparison betweenM1G2M2G2, MM1G2, and SDDG2 of (TM-TCNB)2

The comparison of first ZagrebM1G1, second Zagreb M2G1, second modified Zagreb MM1G1, and symmetry division deg SDDG1 of (TM-TCNB)2 is computationally computed with the assistance of aforesaid M-polynomials given in Table 4. We calculated these indices for different values of z. When z increases, then all the TIs are increased in the same order. We show the graphical representation of all the degree-based TIs in Figure 5 for the particular values of z.Figure 5
Comparison between different Zagreb indices of (TM-TCNB)2.


## 4.3. Comparison betweenHG1, ISG1, and AZIG1 of (TM-TCNB)1

The comparison of the harmonicH, inverse sum (IS), and augmented Zagreb index (AZI) of (TM-TCNB)1 is computationally computed with the assistance of M-polynomial given in Table 5. We calculated these indices for different values of z. When z increases, then all the TIs are increased in the same order. We show the graphical representation of all the degree-based TIs in Figure 6 for the particular values of z.Figure 6
Comparison among different TIs of (TM-TCNB)1.


## 4.4. Comparison betweenHG2, ISG2, and AZIG2 of (TM-TCNB)2

The comparison of the harmonicHG2, inverse sum ISG2, and augmented Zagreb index AZIG2 of (TM-TCNB)2 is computationally computed with the assistance of M-polynomial given in Table 6. We calculated these indices for different values of z. When z increases, then all the TIs are increased in the same order. We show the graphical representation of all the degree-based TIs in Figure 7 for the particular values of z.Figure 7
Comparison among different indices of (TM-TCNB)2.In this work, we discussed the M-polynomials of (TM-TCNB)1 and (TM-TCNB)2. We also work out on the modern degree-based TIs, including 1st Zagreb M1, 2nd Zagreb M2, 2nd modified Zagreb MM2, general Randic RRγ, symmetry division deg (SDD), harmonic H, inverse sum (IS), and the augmented Zagreb index (AZI) and showed their behavior for (TM-TCNB)1 and (TM-TCNB)2. In addition, all the obtained results are also compared with the help of the numerical values provided in the tables, (see Tables 7–10). The problem is still open for the computational results related to different topological indices of the various metal-organic networks.



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