# Multiplicative Attributes Derived from Graph Invariants forSaztec4 Diamond

Authors: Muhammad Haroon Aftab; Imran Siddique; Joshua Kiddy K. Asamoah; Hamiden Abd El-Wahed Khalifa; Muhammad Hussain
Journal: Journal of Mathematics
 (2022)

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## Abstract

This study consists of developing some closed and updated formulas derived from multiplicative graph invariants such as general Randic indexGRIRλ0ϰ for λ0=±1,±1/2, ordinary general geometric-arithmetic (OGA), general version of harmonic index (GHI), sum connectivity index (SI), general sum connectivity index (GSI), 1st and 2nd Gourava and hyper-Gourava indices, (ABC) index, Shegehalli and Kanabur indices, 1st generalised version of Zagreb index (GZI), and forgotten index (FI) for the subdivided Aztec diamond network. Aztec diamond is constructed based on the squares boxes. These square boxes are placed at the centre point and nourish the condition s−1/2+r−1/2≤n. Furthermore, we put a new vertex of degree-2 at each edge of the small boxes, squares in shapes. A new structure is obtained that has the same properties as its parental graph and is called a subdivided Aztec diamond and symbolised as Saztecn. Subsequently, we compute the multiplicative topological attributes to get some new formulas. For this purpose, a simple, connected, and the finite graph is considered by supposing it Y as the graph of the Saztecn. The order and size have also been discussed in this study and found three different kinds of edges (2, 2), (2, 3), and (2, 4) for computing. The discussion on the networks mentioned above provides us with essential results that can be used in the determination of bio and physio activities and can be interspersed with the molecular compounds and their graphical structures better to understand their physical as well as biological properties.


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## Body

## 1. Introduction

The branch of mathematics concerned with graphs, their application, and their correlation with chemical compounds is known as chemical graph theory. In molecular modelling, we use this tool of mathematics. First, the chemical compound’s structure is drawn and compared with its mathematical graphical structure. This theory needs to be used as a mathematical tool to recognise a particular molecular web’s physical and biological features.Vukicevi’c and Furtula’ developed the 1st GA [1] index in 2009 and formulated as(1)GAϒ=∏ro∈Eϒ2dr×dodr+do.An OGA invariant [2] was determined in 2011 and symbolised as given in the following ∀ real w:(2)OGAwϒ=∏ef∈Eϒ4dedfde+dfw.In 2017, V.R. Kulli developed 1st and 2nd Gourava and hyper-Gourava indices [3, 4] are computed by(3)GO1ϒ=∏ef∈Eϒde+df+dedf,GO2ϒ=∏ef∈Eϒde+dfdedf,HGO1ϒ=∏ef∈Eϒde+df+dedf2,HGO2ϒ=∏ef∈Eϒde+dfdedf2.Randic’ index was developed [5] by Milan Randic’ in 1975 and calculated by(4)R1/2ϒ=∏ef∈Eϒ1dedf.Then, Erdos and Bollobas’ invented its general version, familiarised with the general Randic’ index forλo, where λo∈R [6], and evaluated by(5)Rλoϒ=∏ef∈Eϒdedfλo,forλo=−1,1,−12,12.Zhong [7], in 2012, gave the idea of the harmonic index and familiarised by(6)HIϒ=∏ef∈Eϒ2de+df.Yan determined its generalised form [8] in 2015 and symbolised as(7)HwIϒ=∏ef∈Eϒ2de+df.ABC invariant was introduced in 1998 by Estrada et al. [9]; that is,(8)ABCϒ=∏ef∈Eϒde+df−2dedf.SK, SK1, and SK2 invariants [10] were introduced by Shegehalli & Kanabur that are(9)SKϒ=∏ef∈Eϒde+df2,SK1ϒ=∏ef∈Eϒdedf2,SK2ϒ=∏ef∈Eϒde+df22.In 2009, Lucic’ described (SI) [11] and computed it as(10)G−1/2ϒ=∏ef∈Eϒde+df−1/2.Then, in 2010, Zhou and Trinajstic generalised it [8, 12] as(11)Gkϒ=∏ef∈Eϒde+dfk.Zheng, in 2005, developed the general version of the 1st Zagreb index [13](12)M1sϒ=∏ef∈Eϒdes−1+dfs−1,s∈R,s≠0and s≠1.Furtula’ and Gutman [14] formulated an invariant known as F-index(13)Fϒ=∏ef∈Eϒde2+df2


## 2. Material and Methods

Aztec diamond is created based on the square lattices. These square lattices are kept centred at (s, r), satisfying s−1/2+r−1/2≤n. In addition, we place a new node having 2 as the degree at every edge of the small squares. In this way, we get a new structure known as a subdivided Aztec diamond Saztecn. Next, we evaluate the multiplicative topological attributes in order to obtain new formulas. Let us suppose Y is the graph of the Saztecn. The cardinality of Saztecn with respect to vertices is Vϒ=6n2+14n+1, and with respect to edges, is Eϒ=8n2+16n. There are three different kinds of edges (2, 2), (2, 3), and (2, 4).


## 3. Results and Discussion

We have implemented various multiplicative graph invariants [15] over the given molecular structures. Figures 1–4 have been depicted for better understanding. We describe two essential components, nodes and edges. Following, some theorems have been constructed with the help of these particular graphical invariants. Let e=Eϒ be the cardinality of Y with respect to edge set. The Saztecn is established at the terminal nodes of every edge.Figure 1
Saztec (1).Figure 2
Saztec (2).Figure 3
Saztec (3).Figure 4
Saztec (4).Let’s chooseY as a graph of the subdivided Aztec diamond network Saztecn, defining the terms de and df as the degrees of nodes e and f. We have developed the following theorems.Theorem 1.
ForSaztecn, its OGA can be developed as(14)lnOGAwϒ=12wln24​5+w8n2+8n−12ln32​6.Proof.
With the help of Table1, we infer(15)OGAwϒ=∏ef∈Eϒ4de×df​de+dfw,OGAwϒ=16​2+28nw×24​512w×32​6w8n2+8n−12,OGAwϒ=24​512w×32​6w8n2+8n−12.
By making some computations, we get(16)lnOGAwϒ=12wln24​5+w8n2+8n−12ln32​6.Table 1
Describes the partition of edges for graphY.
de,dfforef∈EϒNumber ofEϒ(2, 2)8n(2, 3)12(2, 4)8n2+8n−12Theorem 2.
ForSaztecn, 1st and 2nd Gourava descriptors can be developed as(17)lnGO1ϒ=ln0.05535+8nln112+8n2ln14,lnGO2ϒ=ln3.552×10−3+8nln768+8n2ln48.Proof.
With the help of Table1, we infer(18)GO1ϒ=∏ef∈Eϒde+df+dedf,GO2ϒ=∏ef∈Eϒde+dfdedf.
After some calculations, we have(19)GO1ϒ=88n×1112×148n2+8n−12=0.055351128n×148n2,lnGO1ϒ=ln0.05535+8nln112+8n2ln14,GO2ϒ=168n×3012×488n2+8n−12=3.552×10−37688n×488n2,lnGO2ϒ=ln3.552×10−3+8nln768+8n2ln48.Theorem 3.
ForSaztecn, 1st and 2nd hyper-Gourava descriptors can be developed as(20)lnHGO1ϒ=ln3.0644×10−3+8nln12544+8n2ln196,lnHGO2ϒ=ln1.2621×10−5+8nln589824+8n2ln2304.Proof.
With the help of Table1, we infer(21)HGO1ϒ=∏ef∈Eϒde+df+dedf2,HGO2ϒ=∏ef∈Eϒde+dfdedf2.
By making some computations, we get(22)HGO1ϒ=648n×12112×1968n2+8n−12=3.0644×10−3125448n×1968n2,lnHGO1ϒ=ln3.0644×10−3+8nln12544+8n2ln196,HGO2ϒ=2568n×90012×23048n2+8n−12=1.2621×10−55898248n×23048n2,lnHGO2ϒ=ln1.2621×10−5+8nln589824+8n2ln2304.Theorem 4.
ForSaztecn, GRI can be formulated as(23)Rλ0′Saztecn=ln31.5692−8nln32−8n2ln8,forλ0′=−1,ln0.03703−4nln48−4n2ln12,forλ0′=−12,ln91.125+4nln128+4n2ln8,forλ0′=12,ln0.0316+8nln32+8n2ln8,forλ0′=1.Proof.
We know that(24)Rλ′ϒ=∏ef∈Eϒde×dfλ′,forλ′=±1,±12.Case 1.
Forλ′=−1, its RI can be formulated as(25)R−1ϒ=∏ef∈Eϒ1de×df.
Using (24) and from Table 1, we get(26)R−1ϒ=4−8n×6−12×8−8n2+8n−12.
After some computations, we get(27)lnR−1ϒ=ln31.5692−8nln32−8n2ln8.Case 2.
Forλ′=−1/2, its Randic’ index Rλ′ϒ can be computed as(28)R−1/2ϒ=∏ef∈Eϒ1de×df​.
Using (24) and from Table 1, we know(29)R−1/2ϒ=14​8n×16​12×112​8n2+8n−12.
After some computations, we get(30)lnR−1/2ϒ=ln0.03703−4nln48−4n2ln12.Case 3.
Forλ′=−1/2, its Randic’ index Rλ′ϒ can be computed as(31)R1/2ϒ=∏ef∈Eϒde×df.
Using (24) and from Table 1, getting(32)R1/2ϒ=4​8n×6​12×8​8n2+8n−12.
By doing some calculations, we get(33)lnR1/2ϒ=ln91.125+4nln128+4n2ln8.Case 4.
Forλ′=1, its Randic’ index Rλ′ϒ can be computed as(34)R1ϒ=∏ef∈Eϒde×df1.
Using (24) and from Table 1, we know(35)R1ϒ=48n×612×88n2+8n−12.
By doing some calculations, we get(36)lnR1ϒ=ln0.0316+8nln32+8n2ln8.Theorem 5.
ForSaztecn, HI can be developed as(37)lnHIϒ=ln8.9161−8nln6−8n2ln3.Proof.
With the help of Table1, we infer(38)HIϒ=∏ef∈Eϒ2de+df.
After simplifications, we obtain(39)HIϒ=128n×2512×138n2+8n−12=8.91616−8n×3−8n2,lnHIϒ=ln8.9161−8nln6−8n2ln3.Theorem 6.
ForSaztecn, GHI can be developed as(40)lnHwIϒ=12wln1.2−8nwln6−8wn2ln3.Proof.
With the help of Table1, we infer(41)HwIϒ=∏ef∈Eϒ2de+dfw.
By making some computations, we get(42)HwIϒ=128nw×2512w×138n2+8n−12w=1.212w6−8nw×3−8n2w,lnHwIϒ=12wln1.2−8nwln6−8wn2ln3.Theorem 7.
ForSaztecn, the ABC index can be developed as(43)lnABCϒ=−4nln4−4n2ln2.Proof.
With the help of Table1, we infer(44)ABCϒ=∏ef∈Eϒde+df−2de×df​.
By making some computations, we get(45)ABCϒ=12​8n×12​12×12​8n2+8n−12=4−4n×2−4n2,lnABCϒ=−4nln4−4n2ln2.Theorem 8.
ForSaztecn, SK, SK1, and SK2 descriptors can be developed as(46)lnSKϒ=ln0.8333+8nln6+8n2ln3,lnSK1ϒ=ln0.75+8nln8+8n2ln4,lnSK2ϒ=ln0.6944+8nln36+8n2ln9.Proof.
With the help of Table1, we infer(47)SKϒ=∏ef∈Eϒde+df2,SK1ξ=∏ef∈Eξdedf2,SK2ϒ=∏ef∈Eϒde+df22.
By making some calculations, we get(48)SKϒ=28n×2.512×38n2+8n−12=0.833368n×38n2,lnSKϒ=ln0.8333+8nln6+8n2ln3,SK1ϒ=28n×312×48n2+8n−12=0.7588n×48n2,lnSK1ϒ=ln0.75+8nln8+8n2ln4,SK2ϒ=48n×6.2512×98n2+8n−12=0.6944368n×98n2,lnSK2ϒ=ln0.6944+8nln36+8n2ln9.Theorem 9.
ForSaztecn, SI can be developed as(49)lnSI−1/2ϒ=ln2.9859−4nln24−4n2ln6.Proof.
With the help of Table1, we infer(50)SI−1/2ϒ=∏ef∈Eϒde+df−1/2,SI−1/2ϒ=14​8n×15​12×16​8n2+8n−12.
By making some computations, we get(51)SI−1/2ϒ=2.985924−4n×6−4n2,lnSI−1/2ϒ=ln2.9859−4nln24−4n2ln6.Theorem 10.
ForSaztecn, GSI can be developed as(52)lnGSIwϒ=12wln0.8333+8nwln24+8wn2ln6.Proof.
With the help of Table1, we infer(53)GSIwϒ=∏ef∈Eϒde+dfw,GSIwϒ=48nw×512w×68n2+8n−12w.
By making some computations, we get(54)GSIwϒ=0.833312w248nw×68n2w,lnGSIwϒ=12wln0.8333+8nwln24+8wn2ln6.Theorem 11.
ForSaztecn, 1st GZI can be developed as(55)lnM1wϒ=8nwln2+12ln2w−1+3w−1+8n2+8n−12ln2w−1+4w−1.Proof.
With the help of Table1, we infer(56)M1wϒ=∏ef∈Eϒdew−1+dfw−1,w>1,M1wϒ=28nw×2w−1+3w−112×2w−1+4w−18n2+8n−12,lnM1wϒ=8nwln2+12ln2w−1+3w−1+8n2+8n−12ln2w−1+4w−1.Theorem 12.
ForSaztecn, F-index can be developed as(57)lnFϒ=ln5.688×10−3+8nln160+8n2ln20.Proof.
With the help of Table1, we infer(58)Fϒ=∏ef∈Eϒde2+df2,Fϒ=88n×1312×208n2+8n−12.
By making some computations, we get(59)Fϒ=5.688×10−31608n×208n2,lnFϒ=ln5.688×10−3+8nln160+8n2ln20.


## 4. Main Findings

(i)
Consideration of the molecular structure of the Aztec diamond network(ii)
Getting the subdivided version of the Aztec diamond network forn = 1, 2, 3, 4 by inserting a new node at each edge and placing the name of the new derived molecular structure as Saztec4(iii)
Association of the mathematical graph with the chemical structure(iv)
Vertex labelling for each vertex with their degrees(v)
Edge partition of edge set according to their degrees(vi)
Computations of the degree of each vertex by constructing the generalised formula(vii)
Construction of new closed formulas using many various topological attributes such as general Randic’ indexRλ′ϒ for λ′=±1,±1/2, GHI, OGAI, SHI, GSHI, 1st and 2nd Gourava and hyper-Gourava descriptors, ABC invariant, SKs’ indices, and F-index


## 5. Conclusions

This work involves inventing many new formulas based on multiplicative graph invariants. We have used many indices such as GRI, OGA, GHI, SI, GSI, 1st and 2nd Gourava and hyper-Gourava indices, ABC index, SKs’ indices, 1st GZI and forgotten index (FI) for Saztecn The above-evaluated formulas can be interspersed with the molecular compounds and their graphical structures to understand their physical and biological properties better. More applications can be investigated for these above-mentioned topological indices.



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