# On Reverse Valency Based Topological Characterization of a Chemical Compound

Authors: Maria Singaraj Rosary; Samuel Asefa Fufa
Journal: Journal of Mathematics
 (2022)

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## Abstract

Metal-organic frameworks explicit the consequence of these frameworks with adjustable implementations, namely, energy storage gadgets of magnificent electrode materials, gas store, heterogeneous catalysis, environmental hazard, estimation of chemicals, recognizing of definite gases, controlling solids, and supercapacitors. In this paper, we give explicit expression of the reverse general Randic index, the reverse atom bond connectivity index, the reverse geometric arithmetic index, the reverse forgotten index, the reverse Balaban index, the reverse augmented index, and different types of reverse Zagreb indices of the metal-organic framework M1TPyP-M2 (TPyP = 5, 10, 15, 20-tetrakis (4-pyridyl) porphyrin and M1, M2 = Fe and Co). A graphical comparison of the calculated different types of the reverse degree based topological indices with the aid of the numerical values is also included.


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## Body

## 1. Introduction

Metal-organic frameworks (MOFs) are valuable in modern chemistry. They are described by their three-dimensional frameworks constructed of metal ions and organic molecules. Chu et al. studied the topological indices of MOFs [1]. Certain topological descriptors of certain metal-organic frameworks are well studied in 2020 [2]. In 1959, the initial metal-organic framework was described by Kinoshita et al. [3]. Metal-organic frameworks acquire a great observation required to the work of reticular chemistry for their synthesis [4]. Later, millenarian of MOFs came to be synthesized and widen the range of their prospective applications. Metal-organic frameworks have exhibited applications in the domain of gas catalysis [5], drug delivery [6], and storage [7].LetGVG,XG be an ordered pair of graphs, where VG is a nonempty vertex set and XG is an edge set. A molecular graph is a set of points denoting the atoms in the molecule and collection of lines denoting the covalent bonds. The topological descriptors are helpful in the forecast of physical and chemical properties and the bioactivity of chemical compounds [8, 9]. They are playing a very important role in the field of chemistry.Topological indices are important tools for investigating many physicochemical properties of molecules without performing any testing. They are also used to study Quantitative Structure Activity Relationship (QSAR) of pharmaceuticals to determine their molecular characteristics by numerical computation [10]. Various types of topological indices of graphs are classified into distance-based topological indices, degree-based topological indices, and spectrum-based topological indices. Among these, degree-based topological indices play a vital role in theoretical chemistry and pharmacology. Some important degree-based indices are Randic index, Zagreb indices, Harmonic index, and sum connectivity index. For example, Randic index is one of the excellent molecular descriptors in (QSAR) studies and is desirable for measuring the extent of branching of the carbon-atom skeleton of saturated hydrocarbons [11]. The atom-bond connectivity (ABC) index yields a good representation for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes. For particular physical chemical properties, the diagnostic power of GA index is better than the diagnostic power of the Randić connectivity index [12]. Zagreb-type indices relieved to differentiate some alkane isomers boiling points and have aided in the invention, along with other indices, of a few thousand topological graph indices accepted in the chemical databases [13]. There are a number of research studies on the Zagreb indices and their chemical applications [14].In chemical graph theory, a topological index predicts the biological activity of the molecular graph of chemical compounds [15]. At first, Wiener gave the idea of topological indices during his work on the boiling point of paraffin and mentioned it as “Wiener index,” an initial topological index. In the literature, the most widely used topological indices are Zagreb and Randic [16, 17]. Recently, the topological indices on product graphs and the inverse sum index of hyaluronic acid-paclitaxel molecules applied in anticancer drugs are extensively discussed [18, 19]. Second Zagreb indices of transformation graphs and total transformation graphs and topological indices on hex-derived networks are presented in [20, 21].The degree of vertexp in a graph is the total number of edges incident with vertex p, and it is denoted by χp. A graph can be explained by numerical value, matrix, and polynomial. The concept of reverse vertex degree ℜ⌢p=ΔG−χp+1 was given by Kulli [22]. In a graph, the maximum degree of the vertex is given by ΔG. Degree-, distance-, eccentric-, and spectrum-based indices are special varieties of topological indices. The computation of topological indices has brought up a wide diversity of dormant uses for the evaluation of (QSAR) antiparasitic products. Reverse degree based molecular descriptors of remdesivir used in the treatment of coronavirus are discussed in [23].Recently, the reverse degree based indices of polycyclic metal-organic network were discussed by Zhao et al. [24]. Jung et al. explained the reverse degree based indices of some nanotubes [25]. Reverse degree based molecular descriptors of graphene are discussed in [26]. Reversed degree based topological indices of benzenoid systems are extensively studied in [27]. Recently, Rosary and Liu presented the reverse degree based topological analysis on the line graph and paraline graph of remdesivir used for the treatment of coronavirus [28, 29]. In this article, we examine the physical chemical properties and the biological activity of the molecular structure of the metal-organic framework M1TPyP-M2. Also, we discuss the reverse degree based topological indices of the metal-organic framework M1TPyP-M2 and compare the results graphically.”Milan Randic introduced the first degree based index [17]. Wei et al. defined the reverse Randic index as [23](1)ℜ⌢ℜαG=∑pq∈XGℜ⌢χp×ℜ⌢χqα;α=1,−1,12,−12.Estrada et al. presented the atom bond connectivity index [30]. Wei et al. defined the reverse atom bond connectivity index as [23](2)ℜ⌢ABCG=∑pq∈XGℜ⌢χp+ℜ⌢χq−2ℜ⌢χp×ℜ⌢χq.Vukicevic et al. proposed the geometric arithmetic index [31]. Wei et al. defined the reverse geometric arithmetic index as [23](3)ℜ⌢GAG=∑pq∈XG2ℜ⌢χp×ℜ⌢χqℜ⌢χp+ℜ˜χq.Gutman discussed the first and second Zagreb indices [16, 32]. Wei et al. defined the reverse first and reverse second Zagreb indices as [23](4)ℜ⌢M1G=∑pq∈XGℜ⌢χp+ℜ⌢χq,ℜ⌢M2G=∑pq∈XGℜ⌢χp+ℜ⌢χq.Doslic and Gutman et al. [33, 34] presented the first and second Zagreb co-indices. Wei et al. defined the reverse first and reverse second Zagreb co-indices as [23](5)ℜ⌢M¯1G=2XGVG−1−ℜ⌢M1G,ℜ⌢M¯2G=2XG2−12ℜ⌢M1G−ℜ⌢M2G.Shirdel et al. [35] discussed hyper Zagreb index. Wei et al. defined the reverse hyper Zagreb index as [23](6)ℜ⌢HMG=∑pq∈XGℜ⌢χp+ℜ⌢χq2.Wei et al. defined the reverse first multiple and the reverse second multiple Zagreb indices as [23](7)ℜ⌢PM1G=∏pq∈XGℜ⌢χp+ℜ⌢χq,ℜ⌢PM2G=∏pq∈XGℜ⌢χp+ℜ⌢χq.Furtula and Gutman [36] introduced forgotten index. Wei et al. defined the reverse forgotten index as [23](8)ℜ⌢FG=∑pq∈XGℜ⌢χp2+ℜ⌢χq2.Wei et al. defined the reverse Balban index for a graph of orderq and size p as [23],(9)ℜ⌢JG=pp−q+2∑pq∈XG1ℜ⌢χp×ℜ⌢χq.For a graph, the redefined first, second, and third Zagreb indices are studied in [37]. Wei et al. defined the reverse redefined first, second, and third Zagreb indices as [23](10)ℜ⌢ℜeZG1G=∑pq∈XGℜ⌢χp+ℜ⌢χqℜ⌢χp×ℜ⌢χq,ℜ⌢ℜeZG2G=∑pq∈XGℜ⌢χp×ℜ⌢χqℜ⌢χp+ℜ⌢χq,ℜ⌢ℜeZG3G=∑pq∈XGℜ⌢χp×ℜ⌢χqℜ⌢χp+ℜ⌢χq.Furtula et al. [10] presented augmented Zagreb index. Wei et al. defined the reverse augmented Zagreb index as [23](11)ℜ⌢AZIG=∑pq∈XGℜ⌢χp×ℜ⌢χqℜ⌢χp×ℜ⌢χq−23.


## 2. Main Results

In order to analyse our results, we use the key technique of combinatorial computing, vertices degree counting, edge dividing scheme, logical techniques, and sum of degrees of neighbors’ technique. In addition, we use MATLAB program to do mathematical calculations and plot the results graphically.The 2-dimensional structure of M1TPyP − M2 MOFs is given in Figure1. Let G (p, q) be the graph of M1TPyP − M2 MOFs, where p and q are the number of unit cells in each row and column. The 2-dimensional structure of the molecular graph of G (2, 2) is depicted in Figure 1. The cardinality of vertices and edges in M1TPyP-M2 MOFs is 74pq and 88pq−2p−2q+1.Figure 1
2 × 2 supercell of M1TPyP-M2 MOFs (M1, M2 = Fe and Co).Based on the degree of end vertices, the edges of M1TPyP-M2 MOFs (M1, M2 = Fe and Co) can be partitioned into four partitions. The first edge partition includes of24pq+1 edges pq, where χp=1 and χq=3. The second edge partition includes of 6p+6q−6 edges pq, where χp=2 and χq=3. The third edge partition includes of 56pq−4p−4q+2 edges pq, where χp=3 and χq=3. The fourth edge partition includes 8pq−4p−4q+4 edges pq, where χp=3 and χq=4. Table 1 shows the edge partition of M1TPyP-M2 MOFs.Table 1
Edge partition of M1TPyP-M2 MOFs.
χp,χqNumber of edges(1, 3)24pq+1(2, 3)6p+6q−6(3, 3)56pq−4p−4q+2(3, 4)8pq−4p−4q+4The maximum degree of the vertex of M1TPyP-M2 MOFs is 4. By using the definition of reverse vertex degreeℜ⌢p=ΔG−χp+1, the reverse degree based edge partition of M1TPyP-M2 MOFs is given in Table 2.Table 2
Reverse-based edge partition of M1TPyP-M2 MOFs.
ℜ⌢χp,ℜ⌢χqNumber of edges(4, 2)24pq+1(3, 2)6p+6q−6(2, 2)56pq−4p−4q+2(2, 1)8pq−4p−4q+4Now, we calculate the reverse degree based topological indices as follows.Theorem 1.
We consider the graphGp,q, then the reverse general Randic index is equal to(12)ℜ⌢RαGp,q=416pq+12p+q−20,ifα=1,21pq−2.0004p+q+1.6254,ifα=−1,191.1936pq+1.01364p+q−2.18524,ifα=12,42.14298pq−2.37928p+q+1.732836,ifα=−12.Proof.
We considerGp,q and formula to compute the results(13)ℜ⌢ℜαG=∑pq∈XGℜ⌢χp×ℜ⌢χqα;α=1,−1,12,−12.
Forα=1,(14)ℜ⌢ℜ1G=∑pq∈XGℜ⌢χp×ℜ⌢χq1=24pq+14×2+6p+6q−63×2+56pq−4p−4q+22×2+8pq−4p−4q+42×1=824pq+1+66p+6q−6+456pq−4p−4q+2+28pq−4p−4q+4=192pq+8+36p+36q−36+224pq−16p−16q+8+16pq−8p−8q+8=416pq+12p+q−20.
Forα=−1,(15)ℜ⌢ℜ−1G=∑pq∈XGℜ⌢χp×ℜ⌢χq−1=24pq+14×2+6p+6q−63×2+56pq−4p−4q+22×2+8pq−4p−4q+42×1=24pq+10.125+6p+6q−60.1666+56pq−4p−4q+20.25+8pq−4p−4q+40.5=3pq+0.125+0.9996p+0.9996q−0.9996+14pq−p−q+0.5+4pq−2p−2q+2=21pq−2.0004p+q+1.6254.
Forα=1/2,(16)ℜ⌢ℜ1/2G=∑pq∈XGℜ⌢χp×ℜ⌢χq1/2=24pq+14×2+6p+6q−63×2+56pq−4p−4q+22×2+8pq−4p−4q+42×1=24pq+18+6p+6q−66+56pq−4p−4q+24+8pq−4p−4q+42=24pq+12.8284+6p+6q−62.4494+56pq−4p−4q+22+8pq−4p−4q+41.414=67.8816pq+2.8284+14.66964p+14.66964q−14.66964+112pq−8p−8q+4+11.312pq−5.656p−5.656q+5.656=191.1936pq+1.01364p+q−2.18524.
Forα=−1/2,(17)ℜ⌢ℜ−1/2G=∑pq∈XGℜ⌢χp×ℜ⌢χq−1/2=24pq+14×2+6p+6q−63×2+56pq−4p−4q+22×2+8pq−4p−4q+42×1=24pq+18+6p+6q−66+56pq−4p−4q+24+8pq−4p−4q+42=24pq+12.8284+6p+6q−62.4494+56pq−4p−4q+22+8pq−4p−4q+41.414=24pq+10.353556+6p+6q−60.40826+56pq−4p−4q+20.5+8pq−4p−4q+40.70721=8.4853pq+0.353556+2.44956p+2.44956q−2.44956+28pq−2p−2q+1+5.65768pq−2.82884p−2.82884q+2.82884=42.14298pq−2.37928p+q+1.732836.Theorem 2.
We consider the graphG(p, q), then the reverse Atom Bond Connectivity index is equal to ℜ⌢ABCG=62.2248pq−1.4142p+q+0.7071.Proof.
We considerG(p, q) and formula to compute the results(18)ℜ⌢ABCG=∑pq∈XGℜ⌢χp+ℜ⌢χq−2ℜ⌢χp×ℜ⌢χq=24pq+14+2−24×2+6p+6q−63+2−23×2+56pq−4p−4q+22+2−22×2+8pq−4p−4q+42+1−22×1=24pq+148+6p+6q−636+56pq−4p−4q+212+8pq−4p−4q+412=24pq+10.70710+6p+6q−60.70710+56pq−4p−4q+20.70710+8pq−4p−4q+40.70710=16.9704pq+0.70710+p+q−14.2426+39.5976pq−2.8284p−2.8284q+1.4142+5.6568pq−2.8284p−2.8284q+2.8284=62.2248pq−1.4142p+q+0.7071.Theorem 3.
We consider the graphG (p, q), then the reverse Geometric Arithmetic index is equal to ℜ⌢GAG=86.169672pq−1.892466p+q+0.835266.Proof.
We considerG (p, q) and formula to compute the results(19)ℜ⌢GAG=∑pq∈XG2ℜ⌢χp×ℜ⌢χqℜ⌢χp+ℜ˜χq=24pq+124×24+2+6p+6q−623×23+2+56pq−4p−4q+222×22+2+8pq−4p−4q+422×12+1=24pq+1286+6p+6q−6265+56pq−4p−4q+22×24+8pq−4p−4q+4223=24pq+10.9428+p+q−15.87877+56pq−4p−4q+2+8pq−4p−4q+40.942809=22.6272pq+0.9428+5.87877p+5.87877q−5.87877+56pq−4p−4q+2+7.542472pq−3.771236p−3.771236q+3.771236=86.169672pq−1.892466p+q+0.835266.Theorem 4.
We consider the graphG (p, q), then the reverse First Zagreb index is equal to ℜ⌢M1G=392pq+2p+q−4.Proof.
We considerG(p, q) and formula to compute the results(20)ℜ⌢M1G=∑pq∈XGℜ⌢χp+ℜ⌢χq=24pq+14+2+6p+6q−63+2+56pq−4p−4q+22+2+8pq−4p−4q+42+1=24pq+16+6p+6q−65+56pq−4p−4q+24+8pq−4p−4q+43=144pq+6+30p+30q−30+224pq−16p−16q+8+24pq−12p−12q+12=392pq+2p+q−4.Theorem 5.
We consider the graphG (p, q), then the reverse Second Zagreb index is equal to ℜ⌢M2G=432pq+12p+q−12.Proof.
We considerG(p, q) and formula to compute the results(21)ℜ⌢M2G=∑pq∈XGℜ⌢χp×ℜ⌢χq=24pq+14×2+6p+6q−63×2+56pq−4p−4q+22×2+8pq−4p−4q+42×1=24pq+18+6p+6q−66+56pq−4p−4q+24+8pq−4p−4q+42=192pq+8+36p+36q−36+224pq−16p−16q+8+16pq−8p−8q+8=432pq+12p+q−12.Theorem 6.
We consider the graphG (p, q), then the reverse first Zagreb co-index is equal to ℜ⌢M¯1G=12604pq−296pqp+q+2p+q+2.Proof.
We considerG(p, q) and formula to compute the results(22)ℜ⌢M¯1G=2XGVG−1−ℜ⌢M1G=288pq−2p−2q+174pq−1−392pq+2p+q−4=13024pq−176pq−296p2q+4p−296pq2+4p+148pq−2−392pq−2p−2q+4=12604pq−296pqp+q+2p+q+2.Theorem 7.
We consider the graphG (p, q), then the reverse Second Zagreb Co-index is equal to ℜ⌢M¯2G=15488p2q2−8p2+q2−628pq−13p+q+16.Proof.
We considerG(p, q) and formula to compute the results(23)ℜ⌢M¯2G=2XG2−12ℜ⌢M1G−ℜ⌢M2G=288pq−2p−2q+12−12392pq+2p+q−4−432pq+12p+q−12=15488p2q2−8p2−8q2+2−196pq−p−q+2−432pq−12p−12q+12=15488p2q2−8p2+q2−628pq−13p+q+16.Theorem 8.
We consider the graphG (p, q), then the reverse hyper Zagreb index is equal to ℜ⌢HMG=1832pq+50p+q−46.Proof.
We considerG(p, q) and formula to compute the results(24)ℜ⌢HMG=∑pq∈XGℜ⌢χp+ℜ⌢χq2=24pq+14+22+6p+6q−63+22+56pq−4p−4q+22+22+8pq−4p−4q+42+12=24pq+136+6p+6q−625+56pq−4p−4q+216+8pq−4p−4q+49=864pq+36+150p+150q−150+896pq−64p−64q+32+73pq−36p−36q+36=1832pq+50p+q−46.Theorem 9.
We consider the graphG (p, q), then the reverse forgotten index is equal to ℜ⌢FG=968pq+26p+q−22.Proof.
We considerG(p, q) and formula to compute the results(25)ℜ⌢FG=∑pq∈XGℜ⌢χp2+ℜ⌢χq2=24pq+142+22+6p+6q−632+22+56pq−4p−4q+222+22+8pq−4p−4q+422+12=24pq+120+6p+6q−613+56pq−4p−4q+28+8pq−4p−4q+45=968pq+20+78p+78q−78−32p−32q+16−20p−20q+20=968pq+26p+q−22.Theorem 10.
We consider the graphG (p, q), then the reverse Balaban index is equal to ℜ⌢JG=88pq−2p+q+1/14pq−2p+q+342.14298pq−2.37928p+q+1.732836Proof.
We considerG (p, q) and formula to compute the results
Here,p denotes the size and q denotes the order of graph.(26)ℜ⌢JG=pp−q+2∑pq∈XG1ℜ⌢χp×ℜ⌢χq=88pq−2p−2q+188pq−2p−2q+1−74pq+224pq+14×2+6c+6d−63×2+56pq−4p−4q+22×2+8pq−4p−4q+42×1=88pq−2p−2q+188pq−2p−2q+1−74pq+242.14298pq−2.37928p+q+1.732836=88pq−2p+q+114pq−2p+q+342.14298pq−2.37928p+q+1.732836.Theorem 11.
We consider the graphG(p, q), then the reverse first multiple Zagreb index is equal to(27)ℜ⌢PM1G=1728048p4q28q2−16q+1+2p3672q4−1440q3+844q2−76q+1+p2−768q4+1688q3−1080q2+164q−5+4pq−1212q2−14q+1+p−122q−1.Proof.
We considerG(p, q) and formula to compute the results(28)ℜ⌢PM1G=∏pq∈XGℜ⌢χp+ℜ⌢χq=24pq+14+2+6p+6q−63+2+56pq−4p−4q+22+2+8pq−4p−4q+42+1=24pq+16+6p+6q−65+56pq−4p−4q+24+8pq−4p−4q+43=23224320p4q3−13271040p4q2+829440p4q+23224320p3q4−49766400p3q3+29168640p3q2−2626560p3q+34560p3−13271040p2q4+29168640p2q3−18662400p2q2+2833920p2q−86400p2+829440pq4−2626560pq3+2833920pq2−1105920pq+69120p+34560q3−86400q2+69120q−17280=1728048p4q28q2−16q+1+2p3672q4−1440q3+844q2−76q+1+p2−768q4+1688q3−1080q2+164q−5+4pq−12.12q2−14q+1+p−122q−1.Theorem 12.
We consider the graphG(p, q), then the reverse second multiple Zagreb index is equal to ℜ⌢PM2G=18432[48p4q28q2−16q+1+2p3672q4−1440q3+844q2−76q+1+p2−768q4+1688q3−1080q2+164q−5+4pq−1212q2−14q+1+p−122q−1].Proof.
We considerG(p, q) and formula to compute the results(29)ℜ⌢PM2G=∏pq∈XGℜ⌢χp×ℜ⌢χq=24pq+14×2+6p+6q−63×2+56pq−4p−4q+22×2+8pq−4p−4q+42×1=24pq+18+6p+6q−66+56pq−4p−4q+24+8pq−4p−4q+42=24772608p4q3−14155776p4q2+884736p4q+24772608p3q4−53084160p3q3+31113216p3q2−2801664p3q+34864p3−14155776p2q4+31113216p2q3−19906560p2q2+3022848p2q−92160p2+884736pq4−2801664pq3+3022848pq2−1179648pq+73728p+36864q3−92160q2+73728q−18432=1843248p4q28q2−16q+1+2p3672q4−1440q3+844q2−76q+1+p2−768q4+1688q3−1080q2+164q−5+4pq−12.12q2−14q+1+p−122q−1.Theorem 13.
We consider the graphG(p, q), then the reverse first redefined Zagreb index is equal to ℜ⌢ℜeZG1G=86pq−5p+q+3.75.Proof.
We considerG(p, q) and formula to compute the results(30)ℜ⌢ℜeZG1G=∑pq∈XGℜ⌢χp+ℜ⌢χqℜ⌢χp×ℜ⌢χq=24pq+14+24×2+6p+6q−63+23×2+56pq−4p−4q+22+22×2+8pq−4p−4q+42+12×1=24pq+168+6p+6q−656+56pq−4p−4q+244+8pq−4p−4q+432=86pq−5p+q+3.75.Theorem 14.
We consider the graphG(p, q), then the reverse second redefined Zagreb index is equal to ℜ⌢ℜeZG2G=93.3253pq+0.53336p+q−1.20036.Proof.
We considerG(p, q) and formula to compute the results(31)ℜ⌢ℜeZG2G=∑pq∈XGℜ⌢χp×ℜ⌢χqℜ⌢χp+ℜ⌢χq=24pq+14×24+2+6p+6q−63×23+2+56pq−4p−4q+22×22+2+8pq−4p−4q+42×12+1=24pq+186+6p+6q−665+56pq−4p−4q+244+8pq−4p−4q+423=93.3253pq+0.53336p+q−1.20036.Theorem 15.
We consider the graphG(p, q), then the reverse third redefined Zagreb index is equal to ℜ⌢ℜeZG3G=2096pq+92p+q−76.Proof.
We considerG(p, q) and formula to compute the results(32)ℜ⌢ℜeZG3G=∑pq∈XGℜ⌢χp×ℜ⌢χqℜ⌢χp+ℜ⌢χq=24pq+14×24+2+6p+6q−63×23+2+56pq−4p−4q+22×22+2+8pq−4p−4q+42×12+1=24pq+186+6p+6q−665+56pq−4p−4q+244+8pq−4p−4q+423=2096pq+92p+q−76.Theorem 16.
We consider the graphG(p, q), then the reverse augmented Zagreb index is equal to ℜ⌢AZIG=704pq−16p+q+8.Proof.
We considerG(p, q) and formula to compute the results(33)ℜ⌢AZIG=∑pq∈XGℜ⌢χp×ℜ⌢χqℜ⌢χp×ℜ⌢χq−23=24pq+14×24+2−23+6p+6q−63×23+2−23+56pq−4p−4q+22×22+2−23+8pq−4p−4q+42×12+1−23=24pq+18+p+q−168+56pq−4p−4q+28+8pq−4p−4q+48=704pq−16p+q+8.


## 3. Numerical and Graphical Representation of M1TPYP-M2 MOFS (M1, M2 = Fe and Co)

Here, we compute the numerical results for topological indices for M1TPyP-M2 MOFs to observe the relationship between statistical and biological behaviour. We have applied equal values ofp and q to calculate the reverse degree based indices. Furthermore, we have drawn Figures 2–6 for the structure of M1TPyP-M2 MOFs to examine the graphical behaviour of topological indices calculated above to speculate physiochemical and biological properties. It can be perceived from Tables 3–7 that all topological indices calculated for the structures of M1TPyP-M2 MOFs increase with the increase in p and q. It has been observed clearly from the figures that all indices are in an ascending order as the value of p and q is increasing gradually. Thus, the increasing trend shows that the values of topological indices are increasing accordingly in Tables 3–7.Figure 2
Comparison of the reverse Randic indexℜ⌢ℜ1G,ℜ⌢ℜ−1G,ℜ⌢ℜ1/2G, and ℜ⌢ℜ−1/2G.Figure 3
Comparison of the reverse atom bond connectivity indexℜ⌢ABCG, the reverse geometric arithmetic index ℜ⌢GAG, the reverse first Zagreb index ℜ⌢M1G, and the reverse second Zagreb index ℜ⌢M2G.Figure 4
Graphical comparison of the reverse hyper Zagrebℜ⌢HMG, the reverse first multiple Zagreb index ℜ⌢PM1G, and the reverse second multiple Zagreb index ℜ⌢PM2G.Figure 5
Comparison of the reverse forgotten indexℜ⌢FG, the reverse first Zagreb co-index ℜ⌢M¯1G, and the reverse second Zagreb co-index ℜ⌢M¯2G.Figure 6
Comparison of the reverse Balaban indexℜ⌢JG, the reverse redefined first Zagreb index ℜ⌢ReZG1G, the reverse redefined second Zagreb index ℜ⌢ReZG2G, the reverse redefined third Zagreb index ℜ⌢ReZG3G, and the reverse augmented Zagreb index ℜ⌢AZIG.Table 3
Numerical comparison ofℜ⌢ℜ1G,ℜ⌢ℜ−1G,ℜ⌢ℜ1/2G, and ℜ⌢ℜ−1/2G for M1TPyP-M2 MOFs.
[p, q]ℜ⌢ℜ1Gℜ⌢ℜ−1Gℜ⌢ℜ1/2Gℜ⌢ℜ−1/2G[1, 1]42018.6174191.0356439.117356[2, 2]169277.6094766.643160.7876[3, 3]3796178.6231724.639366.743[4, 4]6732321.62223065.021656.986[5, 5]10500506.62144787.791031.514[6, 6]15100733.62066892.941490.32[7, 7]205321002.61989380.492122033.42[8, 8]267961313.61912250.422660.815[9, 9]338921666.618215502.7413372.487[10, 10]418202061.617419137.4474168.445Table 4
Numerical comparison ofℜ⌢ABCG,ℜ⌢GAG,ℜ⌢M1G, and ℜ⌢M2G for M1TPyP-M2 MOFs.
[p, q]ℜ⌢ABCGℜ⌢GAGℜ⌢M1Gℜ⌢M2G[1, 1]60.103583.220006392444[2, 2]243.949337.9440915721764[3, 3]552.24765.00751835363948[4, 4]984.991364.4102962846996[5, 5]1542.18512136.152406981610908[6, 6]2223.8293080.2331413215684[7, 7]3029.924196.65461923221324[8, 8]3960.465485.41482511627828[9, 9]5015.46036946.5143178435196[10, 10]6194.90318579.95313923643428Table 5
Numerical comparison ofℜ⌢HMG,ℜ⌢PM1G, and ℜ⌢PM2G for M1TPyP-M2 MOFs.
[p, q]ℜ⌢HMGℜ⌢M1Gℜ⌢PM2G[1, 1]1886108000001152000[2, 2]74822.6400e + 092.8159e + 09[3, 3]167425.8740e + 106.2656e + 10[4, 4]296665.0412e + 115.3772e + 11[5, 5]462542.6097e + 122.7837e + 12[6, 6]665069.8791e + 121.0538e + 13[7, 7]904223.0228e + 133.2243e + 13[8, 8]1180027.9277e + 138.4562e + 13[9, 9]1492461.8500e + 141.9733e + 14[10, 10]1841543.9394e + 144.2021e + 14Table 6
Numerical comparison ofℜ⌢FG,ℜ⌢M¯1G, and ℜ⌢M¯2G for M1TPyP-M2 MOFs.
[p, q]ℜ⌢FGℜ⌢M¯1Gℜ⌢M¯2G[1, 1]9981201814834[2, 2]395445690245196[3, 3]8846974661248670[4, 4]156741637943954536[5, 5]244382411229663786[6, 6]3513832589820049124[7, 7]4777441457037154966[8, 8]6234650358663397440[9, 9]78854589394101564386[10, 10]97298668442154815356Table 7
Numerical comparison ofℜ⌢JG,ℜ⌢AZIG,ℜ⌢ReZG1G,ℜ⌢ReZG2G, and ℜ⌢ReZG3G for M1TPyP-M2 MOFs.
[p, q]ℜ⌢JGℜ⌢AZIGℜ⌢ReZG1Gℜ⌢ReZG2Gℜ⌢ReZG3G[1, 1]255.766768079.750093.19172204[2, 2]1.8077e + 032760327.7500374.23438676[3, 3]2.4481e + 036248747.7500841.927519340[4, 4]4.3374e + 03111441.3398e + 031.4963e + 0334196[5, 5]6.7560e + 03174482.1038e + 032.3373e + 0353244[6, 6]9.7041e + 03251603.0398e + 033.3649e + 0376484[7, 7]1.3182e + 04342804.1478e + 034.5792e + 03103916[8, 8]1.7189e + 04448085.4278e + 035.9802e + 03135540[9, 9]2.1727e + 04567446.8798e + 037.5677e + 03171356[10, 10]2.6794e + 04700888.5038e + 039.3420e + 03211364


## 4. Conclusion

In the investigation of the quantitative structure property relationships (QSPRs) and (QSARs), topological descriptors are very important tools to estimate and establish the main characteristics of the bioactivity and chemical compounds. In this article, we have provided the results on reverse degree based topological indices as given in Figures2–6 for M1TPyP-M2 MOFs; furthermore, indices showed increased values for M1TPyP-M2 MOFs. This computational methodology will develop the investigators to recognise the preferred model more easily and would inspire the researchers to focus on the organic framework. The method of computation discussed here is very beneficial to evaluate the physicochemical properties of the stated framework and is cost-effective and time-efficient.



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*Source: 5424430-2022-08-11.xml*