# On Multiplicative Topological Invariants of Magnesium Iodide Structure

Authors: Zhiqiang Zhang; Haidar Ali; Asim Naseem; Usman Babar; Xiujun Zhang; Parvez Ali
Journal: Journal of Mathematics
 (2022)

---

## Abstract

In recent times, the applications of graph theory in molecular and chemical structure research have far exceeded human expectations and have grown exponentially. In this paper, we have determined the multiplicative Zagreb indices, multiplicative hyper-Zagreb indices, multiplicative universal Zagreb indices, sum and product connectivity of multiplicative indices, multiplicative atom-bond connectivity index, and multiplicative geometric-arithmetic index of a famous crystalline structure, magnesium iodideMgI2.


---

## Body

## 1. Introduction and Preliminary Results

In chemical graph theory, molecules can be modeled as graphs by representing atoms as vertices and atomic bonds as edges [1]. The nervous system can be thought of as a graph, where the neurons (nodes) and the edges (bonds) are between them. The nervous system can be seen as a graph, where the nodes are neurons and the edges are the connections between them.The representation of a graph can be expressed by numbers, polynomials, and matrices. Graphs have their own characteristics that may be calculated by topological indices, and under graph automorphism, the topology of graphs remains unchanged. Degree-based topological indices are exceptionally important among different classes of indices and take on a vital role in graph theory and in particular in science [2]. Cheminformatics is a brand-new field that combines chemistry, mathematics, and information science. It investigates quantitative structure-activity (QSAR) and structure-property (QSPR) connections, which are used to predict chemical compound biological activities and qualities.Magnesium iodide (MgI2) is a chemical compound and has many commercial uses. It is used for manufacture of different organic compounds in large scale due to its cheapness and availability everywhere in the world. Magnesium iodide can be obtained from the reaction of hydro-iodic acid with magnesium oxide, magnesium hydroxide, and magnesium carbonate.(1)2HI+MgCO3⟶MgI2+H2O+CO22HI+MgO⟶MgI2+H2O2HI+MgOH2⟶MgI2+2H2OEvery heptagon inMgI2 is connected to each other in column and row by making three C4 within each heptagon. We will use “m” to represent the number of C4 rows on the upper side and “n” to represent the number of C4 on the lower side of the heptagon.In Figures1 and 2, there are variables m and n, which are dependent on each other to maintain the structure of magnesium iodide. Magnesium has 2 electrons in its outermost shell, while iodide contains 7 electrons in its valence shell. As a result, to turn out to be stable, magnesium offers its electrons to iodide. To maintain the chemical structure of magnesium iodide, it depends on each other. As a result, the graph was subdivided into even and odd vertices. For the first type, the cardinality of vertices and edges of graph L1, as shown in Figure 1, is m=2n+1, and for the second type, the cardinality of vertices and edges of graph L2, as shown in Figure 2, is m=2n+2, where n∈ℕ, respectively.Figure 1
GraphL1 of MgI2 for m=2n+1.Figure 2
GraphL2 of MgI2 for m=2n+2.In this article, “L” is considered a structure with a vertex set VL and edge set EL, and dr is the degree of vertex r∈VL.The “second multiplicative Zagreb index” [4] is defined as(2)II2L=∏rs∈ELdr×ds,wherer,s is any unordered pair of vertices in L.Kulli [5] further described some new and advanced topological indices, and he named them as the “first multiplicative hyper-Zagreb index” and the “second multiplicative hyper-Zagreb index” of a graph “L.” The indices are defined as(3)HII1L=∏rs∈ELdr+ds2,(4)HII2L=∏rs∈ELdr×ds2.The “first multiplicative universal Zagreb index” and the “second multiplicative universal Zagreb index” were introduced by Kulli et al. [6]. These indices are defined as(5)MZ1aL=∏rs∈ELdr+dsa,(6)MZ2aL=∏rs∈ELdr×dsa,where a∈ℕ.The multiplicative sum and product connectivity indices were introduced by Kulli [7], defined as(7)SCIIL=∏rs∈EL1dr+ds,(8)PCIIL=∏rs∈EL1dr×ds.“Multiplicative atom-bond connectivity index,” “multiplicative geometric-arithmetic index,” and “multiplicative universal geometric-arithmetic index” for a simple graphL were introduced by Kulli [7], and these indices are defined as(9)ABCIIL=∏rs∈ELdr+ds−2dr×ds,(10)GAIIL=∏rs∈EL2dr×dsdr+ds,(11)GAaIIL=∏rs∈EL2dr×dsdr+dsa,where a∈ℕ.


## 2. Main Results

Afzal et al. [8] computed the molecular description for magnesium iodide. In this article, we shall discuss the first type and second type of magnesium iodide structure and compute the exact results for topological indices like multiplicative Zagreb indices, multiplicative hyper-Zagreb indices, multiplicative universal Zagreb indices, sum and product connectivity of multiplicative indices, multiplicative atom-bond connectivity index, and multiplicative geometric-arithmetic index for the magnesium iodide structure. For further study of topological indices of various graph families, see [9–14]. For basic definitions and notations, see [15, 16].


### 2.1. Results for the First Type of Magnesium Iodide Structure

In this section, we compute topological indices such as multiplicative Zagreb indices, multiplicative hyper-Zagreb indices, multiplicative universal Zagreb, sum and product connectivity of multiplicative indices, multiplicative atom-bond connectivity index, and multiplicative geometric and universal geometric-arithmetic indices for the first type of magnesium iodide structureL1, as shown in Figure 1.Theorem 1.
Consider the magnesium iodide structureMgI2 of first type; then, the second multiplicative Zagreb index is equal to(12)II2L=13931406950400nn+52n+827n−13.Proof.
LetL1 be the magnesium iodide structure MgI2 for m=2n+1,n∈ℕ of first type. In Table 1, there is an edge partition of L1. From equation (2),(13)II2L1=∏rs∈EL1dr×ds.
By applying edge partition of Table1,(14)II2L1=3E1L1×4E2L1×6E3L1×10E4L1×12E5L1×6E6L1×9E7L1×12E8L1×15E9L1×18E10L1×8E11L1=31×41×6n+5×108×122n+8×62×93n×121×1512×1827n−13×82.
We obtain the following by making some calculations:(15)⇒II2L1=13931406950400nn+52n+827n−13.Table 1
Edge subdivision ofL1 of MgI2 based on degree of end vertices of each edge for m=2n+1, where n∈ℕ.
dr,ds where rs∈EL1Number of edges(1, 3)1(1, 4)1(1, 6)n+5(2, 5)8(2, 6)2n+8(3, 2)2(3, 3)3n(3, 4)1(3, 5)12(3, 6)27n−13(4, 2)2Theorem 2.
Consider the magnesium iodide structureMgI2 of first type; then, the first and second multiplicative hyper-Zagreb indices are equal to(16)HII1L1=582761278687150080000nn+52n+827n−13,HII2L1=168475780918101934080000nn+52n+827n−13.Proof.
LetL1 be the magnesium iodide structure MgI2 of first type. In Table 1, there is an edge partition of L1. From equation (3),(17)HII1L1=∏rs∈EL1dr+ds2.
By applying edge partition of Table1,(18)HII1L1=16E1L1×25E2L1×49E3L1×49E4L1×64E5L1×25E6L1×36E7L1×49E8L1×64E9L1×81E10L1×36E11L1=161×251×49n+5×498×642n+8×252×363n×491×6412×8127n−13×362.
We obtain the following by making some calculations:(19)⇒HII1L1=582761278687150080000nn+52n+827n−13.
From equation (4),(20)HII2L1=∏rs∈EL1dr×ds2.
By applying edge partition of Table1,(21)HII2L1=9E1L1×16E2L1×36E3L1×100E4L1×144E5L1×36E6L1×81E7L1×144E8L1×225E9L1×324E10L1×64E11L1=91×161×36n+5×1008×1442n+8×362×813n×1441×22512×32427n−13×642.
We obtain the following by making some calculations:(22)⇒HII2L1=168475780918101934080000nn+52n+827n−13.Theorem 3.
Consider the magnesium iodide structureMgI2 of first type; then, the first and second multiplicative universal Zagreb indices are equal to(23)MZ1aL1=27+10a×32+4a×52a×73ann+52n+827n−13,MZ2aL1=27+13a×32+10a×52a×nn+52n+827n−13.Proof.
LetL1 be the magnesium iodide structure MgI2 of first type. In Table 1, there is an edge partition of L1. From equation (5),(24)MZ1aL1=∏rs∈EL1dr+dsa.
By applying edge partition of Table1,(25)MZ1aL1=4aE1L1×5aE2L1×7aE3L1×7aE4L1×8aE5L1×5aE6L1×6aE7L1×7aE8L1×8aE9L1×9aE10L1×6aE11L1=4a1×5a1×7an+5×7a8×8a2n+8×5a2×6a3n×7a1×8a12×9a27n−13×6a2.
We obtain the following by making some calculations:(26)⇒MZ1aL1=27+10a×32+4a×52a×73ann+52n+827n−13.
From equation (6),(27)MZ2aL1=∏rs∈EL1dr×dsa.
By applying edge partition of Table1,(28)MZ1aL1=3aE1L1×4aE2L1×6aE3L1×10aE4L1×12aE5L1×6aE6L1×9aE7L1×12aE8L1×15aE9L1×18aE10L1×8aE11L1=3a1×4a1×6an+5×10a8×12a2n+8×6a2×9a3n×12a1×15a12×18a27n−13×8a2.
We obtain the following by making some calculations:(29)⇒MZ2aL1=27+13a×32+10a×52a×nn+52n+827n−13.Theorem 4.
Consider the magnesium iodide structureMgI2 of first type; then, the multiplicative sum and product connectivity indices are equal to(30)SCIIL1=4357nn+52n+827n−13,PCIIL1=2135nn+52n+827n−13.Proof.
LetL1 be the magnesium iodide structure MgI2 of first type. In Table 1, there is an edge partition of L1. From equation (7),(31)SCIIL1=∏rs∈Eϒ1dr+ds.
By applying edge partition of Table1,(32)SCIIL1=14E1L1×15E2L1×17E3L1×17E4L1×18E5L1×15E6L1×16E7L1×17E8L1×18E9L1×19E10L1×16E11L1=141×151×17n+5×178×182n+8×152×163n×171×1812×199×162.
We obtain the following by making some calculations:(33)⇒SCIIL1=4357nn+52n+827n−13.
From equation (8),(34)PCIIL1=∏rs∈EL11dr×ds.
By applying edge partition of Table1,(35)PCIIL1=13E1L1×14E2L1×16E3L1×110E4L1×112E5L1×16E6L1×19E7L1×112E8L1×115E9L1×118E10L1×18E11L1=131×141×16n+5×1108×1122n+8×162×193n×1121×11512×1189×182.
We obtain the following by making some calculations:(36)⇒PCIIL1=2135nn+52n+827n−13.Theorem 5.
Consider the magnesium iodide structureMgI2 of first type; then, the multiplicative atom-bond connectivity and geometric-arithmetic indices are equal to(37)ABCIIL1=16353nn+52n+827n−13,GAIIL1=44236821715nn+52n+827n−13.Proof.
LetL1 be the magnesium iodide structure MgI2 of first type. In Table 1, there is an edge partition of L1. From equation (9),(38)ABCIIL1=∏rs∈EMgI2dr+ds−2dr×ds.
By applying edge partition of Table1,(39)ABCIIL1=23E1L1×34E2L1×56E3L1×510E4L1×612E5L1×36E6L1×49E7L1×512E8L1×615E9L1×718E10L1×48E11L1=231×341×56n+5×5108×6122n+8×362493n×1512×61512×71827n−13×482.
We obtain the following by making some calculations:(40)⇒ABCIIL1=16353nn+52n+827n−13.
From equation (10),(41)GAIIL1=∏rs∈EL12dr×dsdr+ds.
By applying edge partition of Table1,(42)GAIIL1=234E1L1×245E2L1×267E3L1×2107E4L1×2128E5L1×265E6L1296E7L1×2127E8L1×2158E9L1×2189E10L1×286E11L1=2341×2451×267n+5×21078×21282n+8×2652×2963n×21271×215812×218927n−13×2862.
We obtain the following by making some calculations:(43)⇒GAIIL1=44236821715nn+52n+827n−13.Theorem 6.
Consider the magnesium iodide structureMgI2 of first type; then, the universal multiplicative geometric-arithmetic index is equal to(44)GAaIIL1=214+15a/2×32+a×5−a×7−3ann+52n+827n−13.Proof.
LetL1 be the magnesium iodide structure MgI2 of first type. In Table 1, there is an edge partition of L1. From equation (10),(45)GAaIIL1=∏rs∈EL12dr×dsdr+dsa.
By applying edge partition of Table1,(46)GAaIIL1=234aE1L1×245aE2L1×267aE3L1×2107aE4L1×E5L12128a×265aE6L1×296aE7L1×2127aE8L1×2158aE9L1×2189aE10L1×286aE11L1=234a1×2451a×267an+5×21078a×2128a2n+8×265a2×296a3n×2127a1×2158a12×2189a27n−13×286a2.
We obtain the following by making some calculations:(47)⇒GAaIIL1=214+15a/2×32+a×5−a×7−3ann+52n+827n−13.


### 2.2. Results for the Second Type of Magnesium Iodide Structure

In this section, we compute topological indices such as multiplicative Zagreb indices, multiplicative hyper-Zagreb indices, multiplicative universal Zagreb indices, sum and product connectivity of multiplicative indices, multiplicative atom-bond connectivity index, and multiplicative geometric and universal geometric-arithmetic indices for the second type of magnesium iodide structure.Theorem 7.
Consider the magnesium iodide structureMgI2 of second type; then, the second multiplicative Zagreb index is equal to(48)II2L2=75582720000n+52n+83n+127n+7.Proof.
LetL2 be the magnesium iodide structure MgI2 of second type. In Table 2, there is an edge partition of L2. From equation (2),(49)II2L2=∏rs∈EL2dr×ds.
By applying edge partition of Table2,(50)II2L2=3E1L2×5E2L2×6E3L2×4E4L2×10E5L2×12E6L2×6E7L2×9E8L2×15E9L2×18E10L2=31×51×6n+5×45×102×122n+8×66×93n+1×152×1827n+7.
We obtain the following by making some calculations:(51)⇒II2L2=75582720000n+52n+83n+127n+7.Table 2
Edge subdivision ofL2MgI2 based on degree of end vertices of each edge for m=2n+2, where n∈ℕ.
dr,ds where rs∈EL2Number of edges(1, 3)1(1, 5)1(1, 6)n+5(2, 2)5(2, 5)2(2, 6)2n+8(3, 2)6(3, 3)3n+1(3, 5)2(3, 6)27n+7Theorem 8.
Consider the magnesium iodide structureMgI2 of second type; then, the first and second multiplicative hyper-Zagreb indices are equal to(52)HII1L2=792872488009728000n+52n+83n+127n+7,HII2L2=47606229688320000000n+52n+83n+127n+7.Proof.
LetL2 be the magnesium iodide structure MgI2 of second type. In Table 2, there is an edge partition of L2. From equation (3),(53)HII1L2=∏rs∈EL2dr+ds2.
By applying edge partition of Table2,(54)HII1L2=16E1L2×36E2L2×49E3L2×16E4L2×49E5L2×64E6L2×25E7L2×36E8L2×64E9L2×81E10L2=161×361×49n+5×165×492×642n+8×256×363n+1×642×8127n+7.
We obtain the following by making some calculations:(55)⇒HII1L2=792872488009728000n+52n+83n+127n+7.
From equation (4),(56)HII2L2=∏rs∈EL2dr×ds2.
By applying edge partition of Table2,(57)HII2L2=9E1L2×25E21L2×36E3L2×16E4L2×100E5L2×144E6L2×36E7L2×81E8L2×225E9L2×324E10L2,=91×251×36n+5×165×1002×1442n+8×366×813n+1×2252×32427n+7.
We obtain the following by making some calculations:(58)⇒HII2L2=47606229688320000000n+52n+83n+127n+7.Theorem 9.
Consider the magnesium iodide structureMgI2 of second type; then, the first and second multiplicative universal Zagreb indices are equal to(59)MZ1aL2=51+a×72a×241+4an+52n+83n+127n+7,MZ2aL2=23+8a×31+9a×51+3a×n+52n+83n+127n+7.Proof.
LetL2 be the magnesium iodide structure MgI2 of second type. In Table 2, there is an edge partition of L2. From equation (5),(60)MZ1aL2=∏rs∈EL2dr+dsa.
By applying edge partition of Table2,(61)MZ1aL2=4aE1L2×6aE2L2×7aE3L2×4aE4L2×7aE5L2×8aE6L2×5aE7L2×6aE8L2×8aE9L2×9aE10L2=4a1×6a1×7an+5×4a5×7a2×8a2n+8×5a6×6a3n+1×8a2×9a27n+7.
We obtain the following by making some calculations:(62)⇒MZ1aL2=51+a×72a×241+4an+52n+83n+127n+7.
From equation (6),(63)MZ2aL2=∏rs∈EL2dr×dsa.
By applying edge partition of Table2,(64)MZ2aL2=3aE1L2×5aE2L2×6aE3L2×4aE4L2×10aE5L2×12aE6L2×6aE7L2×9aE8L2×15aE9L2×18aE10L2=3a1×5a1×6an+5×4a5×10a2×12a2n+8×6a6×9a3n+1×15a2×18a27n+7.
We obtain the following by making some calculations:(65)⇒MZ2aL2=23+8a×31+9a×51+3a×n+52n+83n+127n+7.Theorem 10.
Consider the magnesium iodide structureMgI2 of second type; then, the multiplicative sum and product connectivity indices are equal to(66)SCIIL2=11685n+52n+83n+127n+7,PCIIL2=15415n+52n+83n+127n+7.Proof.
LetL2 be the magnesium iodide structure MgI2 of second type. In Table 2, there is an edge partition of L2. From equation (7),(67)SCIIL2=∏rs∈EL21dr+ds.
By applying edge partition of Table2,(68)SCIIL2=14E1L2×16E2L2×17E3L2×14E4L2×17E5L2×18E6L2×15E7L2×16E8L2×18E9L2×19E10L2=141×161×17n+5×145×172×182n+8×156×163n+1×182×1927n+7.
We obtain the following by making some calculations:(69)⇒SCIIL2=11685n+52n+83n+127n+7.
From equation (8),(70)PCIIL2=∏rs∈EL21dr×ds.
By applying edge partition of Table2,(71)PCIIL2=13E1L2×15E2L2×16E3L2×14E4L2×110E5L2×112E6L2×16E7L2×19E7L2×115E9L2×118E10L2=131×151×16n+5×145×1102×1122n+8×166×193n+1×1152×11827n+7.
We obtain the following by making some calculations:(72)⇒PCIIL2=15415n+52n+83n+127n+7.Theorem 11.
Consider the magnesium iodide structureMgI2 of second type; then, the multiplicative atom-bond connectivity index and geometric-arithmetic index are equal to(73)ABCIIL2=8359n+52n+83n+127n+7,GAIIL2=4801549n+52n+83n+127n+7.Proof.
LetL2 be the magnesium iodide structure MgI2 of second type. In Table 2, there is an edge partition of L2. From equation (9),(74)ABCIIL2=∏rs∈EL2dr+ds−2dr×ds.
By applying edge partition of Table2,(75)ABCIIL2=23E1L2×45E2L2×56E3L2×24E4L2×510E5L2×612E6L236E7L2×46E8L2×615E9L2×718E10L2=231×451×56n+5×245×5102×6122n+8366×3n+149×6152×71827n+7.
We obtain the following by making some calculations:(76)⇒ABCIIL2=8359n+52n+83n+127n+7.
From equation (9),(77)GAIIL2=∏rs∈EL22dr×dsdr+ds.
By applying edge partition of Table2,(78)GAIIL2=234E1L2×256E2L2×267E3L2×244E4L2×2107E5L2×2128E6L2×265E7L2×296E8L2×2158E9L2×2189E10L2=2341×2561×267n+5×2445×21072×21282n+82656×2963n+1×21582×218927n+7.
We obtain the following by making some calculations:(79)⇒GAIIL2=4801549n+52n+83n+127n+7.Theorem 12.
Consider the magnesium iodide structureMgI2 of second type; then, the universal multiplicative geometric-arithmetic index is equal to(80)GAaIIL2=23+2a×7−2a×152+a/2n+52n+83n+127n+7.Proof.
LetL2 be the magnesium iodide structure MgI2 of second type. In Table 2, there is an edge partition of L2. From equation (10),(81)GAaIIL2=∏rs∈EL22dr×dsdr+dsa.
By applying edge partition of Table2,(82)GAaIIL2=234aE1L2×256aE2L2×267aE3L2×244aE4L2×2107aE5L2×2128aE6L2×265aE7L2×296aE8L2×2158aE9L2×2189aE10L2=234a1×256a1×267an+5×244a5×2107a2×2128a2n+8265a6×296a3n+1×2158a2×2189a27n+7.
We obtain the following by making some calculations:(83)⇒GAaIIL2=23+2a×7−2a×152+a/2n+52n+83n+127n+7.


## 2.1. Results for the First Type of Magnesium Iodide Structure

In this section, we compute topological indices such as multiplicative Zagreb indices, multiplicative hyper-Zagreb indices, multiplicative universal Zagreb, sum and product connectivity of multiplicative indices, multiplicative atom-bond connectivity index, and multiplicative geometric and universal geometric-arithmetic indices for the first type of magnesium iodide structureL1, as shown in Figure 1.Theorem 1.
Consider the magnesium iodide structureMgI2 of first type; then, the second multiplicative Zagreb index is equal to(12)II2L=13931406950400nn+52n+827n−13.Proof.
LetL1 be the magnesium iodide structure MgI2 for m=2n+1,n∈ℕ of first type. In Table 1, there is an edge partition of L1. From equation (2),(13)II2L1=∏rs∈EL1dr×ds.
By applying edge partition of Table1,(14)II2L1=3E1L1×4E2L1×6E3L1×10E4L1×12E5L1×6E6L1×9E7L1×12E8L1×15E9L1×18E10L1×8E11L1=31×41×6n+5×108×122n+8×62×93n×121×1512×1827n−13×82.
We obtain the following by making some calculations:(15)⇒II2L1=13931406950400nn+52n+827n−13.Table 1
Edge subdivision ofL1 of MgI2 based on degree of end vertices of each edge for m=2n+1, where n∈ℕ.
dr,ds where rs∈EL1Number of edges(1, 3)1(1, 4)1(1, 6)n+5(2, 5)8(2, 6)2n+8(3, 2)2(3, 3)3n(3, 4)1(3, 5)12(3, 6)27n−13(4, 2)2Theorem 2.
Consider the magnesium iodide structureMgI2 of first type; then, the first and second multiplicative hyper-Zagreb indices are equal to(16)HII1L1=582761278687150080000nn+52n+827n−13,HII2L1=168475780918101934080000nn+52n+827n−13.Proof.
LetL1 be the magnesium iodide structure MgI2 of first type. In Table 1, there is an edge partition of L1. From equation (3),(17)HII1L1=∏rs∈EL1dr+ds2.
By applying edge partition of Table1,(18)HII1L1=16E1L1×25E2L1×49E3L1×49E4L1×64E5L1×25E6L1×36E7L1×49E8L1×64E9L1×81E10L1×36E11L1=161×251×49n+5×498×642n+8×252×363n×491×6412×8127n−13×362.
We obtain the following by making some calculations:(19)⇒HII1L1=582761278687150080000nn+52n+827n−13.
From equation (4),(20)HII2L1=∏rs∈EL1dr×ds2.
By applying edge partition of Table1,(21)HII2L1=9E1L1×16E2L1×36E3L1×100E4L1×144E5L1×36E6L1×81E7L1×144E8L1×225E9L1×324E10L1×64E11L1=91×161×36n+5×1008×1442n+8×362×813n×1441×22512×32427n−13×642.
We obtain the following by making some calculations:(22)⇒HII2L1=168475780918101934080000nn+52n+827n−13.Theorem 3.
Consider the magnesium iodide structureMgI2 of first type; then, the first and second multiplicative universal Zagreb indices are equal to(23)MZ1aL1=27+10a×32+4a×52a×73ann+52n+827n−13,MZ2aL1=27+13a×32+10a×52a×nn+52n+827n−13.Proof.
LetL1 be the magnesium iodide structure MgI2 of first type. In Table 1, there is an edge partition of L1. From equation (5),(24)MZ1aL1=∏rs∈EL1dr+dsa.
By applying edge partition of Table1,(25)MZ1aL1=4aE1L1×5aE2L1×7aE3L1×7aE4L1×8aE5L1×5aE6L1×6aE7L1×7aE8L1×8aE9L1×9aE10L1×6aE11L1=4a1×5a1×7an+5×7a8×8a2n+8×5a2×6a3n×7a1×8a12×9a27n−13×6a2.
We obtain the following by making some calculations:(26)⇒MZ1aL1=27+10a×32+4a×52a×73ann+52n+827n−13.
From equation (6),(27)MZ2aL1=∏rs∈EL1dr×dsa.
By applying edge partition of Table1,(28)MZ1aL1=3aE1L1×4aE2L1×6aE3L1×10aE4L1×12aE5L1×6aE6L1×9aE7L1×12aE8L1×15aE9L1×18aE10L1×8aE11L1=3a1×4a1×6an+5×10a8×12a2n+8×6a2×9a3n×12a1×15a12×18a27n−13×8a2.
We obtain the following by making some calculations:(29)⇒MZ2aL1=27+13a×32+10a×52a×nn+52n+827n−13.Theorem 4.
Consider the magnesium iodide structureMgI2 of first type; then, the multiplicative sum and product connectivity indices are equal to(30)SCIIL1=4357nn+52n+827n−13,PCIIL1=2135nn+52n+827n−13.Proof.
LetL1 be the magnesium iodide structure MgI2 of first type. In Table 1, there is an edge partition of L1. From equation (7),(31)SCIIL1=∏rs∈Eϒ1dr+ds.
By applying edge partition of Table1,(32)SCIIL1=14E1L1×15E2L1×17E3L1×17E4L1×18E5L1×15E6L1×16E7L1×17E8L1×18E9L1×19E10L1×16E11L1=141×151×17n+5×178×182n+8×152×163n×171×1812×199×162.
We obtain the following by making some calculations:(33)⇒SCIIL1=4357nn+52n+827n−13.
From equation (8),(34)PCIIL1=∏rs∈EL11dr×ds.
By applying edge partition of Table1,(35)PCIIL1=13E1L1×14E2L1×16E3L1×110E4L1×112E5L1×16E6L1×19E7L1×112E8L1×115E9L1×118E10L1×18E11L1=131×141×16n+5×1108×1122n+8×162×193n×1121×11512×1189×182.
We obtain the following by making some calculations:(36)⇒PCIIL1=2135nn+52n+827n−13.Theorem 5.
Consider the magnesium iodide structureMgI2 of first type; then, the multiplicative atom-bond connectivity and geometric-arithmetic indices are equal to(37)ABCIIL1=16353nn+52n+827n−13,GAIIL1=44236821715nn+52n+827n−13.Proof.
LetL1 be the magnesium iodide structure MgI2 of first type. In Table 1, there is an edge partition of L1. From equation (9),(38)ABCIIL1=∏rs∈EMgI2dr+ds−2dr×ds.
By applying edge partition of Table1,(39)ABCIIL1=23E1L1×34E2L1×56E3L1×510E4L1×612E5L1×36E6L1×49E7L1×512E8L1×615E9L1×718E10L1×48E11L1=231×341×56n+5×5108×6122n+8×362493n×1512×61512×71827n−13×482.
We obtain the following by making some calculations:(40)⇒ABCIIL1=16353nn+52n+827n−13.
From equation (10),(41)GAIIL1=∏rs∈EL12dr×dsdr+ds.
By applying edge partition of Table1,(42)GAIIL1=234E1L1×245E2L1×267E3L1×2107E4L1×2128E5L1×265E6L1296E7L1×2127E8L1×2158E9L1×2189E10L1×286E11L1=2341×2451×267n+5×21078×21282n+8×2652×2963n×21271×215812×218927n−13×2862.
We obtain the following by making some calculations:(43)⇒GAIIL1=44236821715nn+52n+827n−13.Theorem 6.
Consider the magnesium iodide structureMgI2 of first type; then, the universal multiplicative geometric-arithmetic index is equal to(44)GAaIIL1=214+15a/2×32+a×5−a×7−3ann+52n+827n−13.Proof.
LetL1 be the magnesium iodide structure MgI2 of first type. In Table 1, there is an edge partition of L1. From equation (10),(45)GAaIIL1=∏rs∈EL12dr×dsdr+dsa.
By applying edge partition of Table1,(46)GAaIIL1=234aE1L1×245aE2L1×267aE3L1×2107aE4L1×E5L12128a×265aE6L1×296aE7L1×2127aE8L1×2158aE9L1×2189aE10L1×286aE11L1=234a1×2451a×267an+5×21078a×2128a2n+8×265a2×296a3n×2127a1×2158a12×2189a27n−13×286a2.
We obtain the following by making some calculations:(47)⇒GAaIIL1=214+15a/2×32+a×5−a×7−3ann+52n+827n−13.


## 2.2. Results for the Second Type of Magnesium Iodide Structure

In this section, we compute topological indices such as multiplicative Zagreb indices, multiplicative hyper-Zagreb indices, multiplicative universal Zagreb indices, sum and product connectivity of multiplicative indices, multiplicative atom-bond connectivity index, and multiplicative geometric and universal geometric-arithmetic indices for the second type of magnesium iodide structure.Theorem 7.
Consider the magnesium iodide structureMgI2 of second type; then, the second multiplicative Zagreb index is equal to(48)II2L2=75582720000n+52n+83n+127n+7.Proof.
LetL2 be the magnesium iodide structure MgI2 of second type. In Table 2, there is an edge partition of L2. From equation (2),(49)II2L2=∏rs∈EL2dr×ds.
By applying edge partition of Table2,(50)II2L2=3E1L2×5E2L2×6E3L2×4E4L2×10E5L2×12E6L2×6E7L2×9E8L2×15E9L2×18E10L2=31×51×6n+5×45×102×122n+8×66×93n+1×152×1827n+7.
We obtain the following by making some calculations:(51)⇒II2L2=75582720000n+52n+83n+127n+7.Table 2
Edge subdivision ofL2MgI2 based on degree of end vertices of each edge for m=2n+2, where n∈ℕ.
dr,ds where rs∈EL2Number of edges(1, 3)1(1, 5)1(1, 6)n+5(2, 2)5(2, 5)2(2, 6)2n+8(3, 2)6(3, 3)3n+1(3, 5)2(3, 6)27n+7Theorem 8.
Consider the magnesium iodide structureMgI2 of second type; then, the first and second multiplicative hyper-Zagreb indices are equal to(52)HII1L2=792872488009728000n+52n+83n+127n+7,HII2L2=47606229688320000000n+52n+83n+127n+7.Proof.
LetL2 be the magnesium iodide structure MgI2 of second type. In Table 2, there is an edge partition of L2. From equation (3),(53)HII1L2=∏rs∈EL2dr+ds2.
By applying edge partition of Table2,(54)HII1L2=16E1L2×36E2L2×49E3L2×16E4L2×49E5L2×64E6L2×25E7L2×36E8L2×64E9L2×81E10L2=161×361×49n+5×165×492×642n+8×256×363n+1×642×8127n+7.
We obtain the following by making some calculations:(55)⇒HII1L2=792872488009728000n+52n+83n+127n+7.
From equation (4),(56)HII2L2=∏rs∈EL2dr×ds2.
By applying edge partition of Table2,(57)HII2L2=9E1L2×25E21L2×36E3L2×16E4L2×100E5L2×144E6L2×36E7L2×81E8L2×225E9L2×324E10L2,=91×251×36n+5×165×1002×1442n+8×366×813n+1×2252×32427n+7.
We obtain the following by making some calculations:(58)⇒HII2L2=47606229688320000000n+52n+83n+127n+7.Theorem 9.
Consider the magnesium iodide structureMgI2 of second type; then, the first and second multiplicative universal Zagreb indices are equal to(59)MZ1aL2=51+a×72a×241+4an+52n+83n+127n+7,MZ2aL2=23+8a×31+9a×51+3a×n+52n+83n+127n+7.Proof.
LetL2 be the magnesium iodide structure MgI2 of second type. In Table 2, there is an edge partition of L2. From equation (5),(60)MZ1aL2=∏rs∈EL2dr+dsa.
By applying edge partition of Table2,(61)MZ1aL2=4aE1L2×6aE2L2×7aE3L2×4aE4L2×7aE5L2×8aE6L2×5aE7L2×6aE8L2×8aE9L2×9aE10L2=4a1×6a1×7an+5×4a5×7a2×8a2n+8×5a6×6a3n+1×8a2×9a27n+7.
We obtain the following by making some calculations:(62)⇒MZ1aL2=51+a×72a×241+4an+52n+83n+127n+7.
From equation (6),(63)MZ2aL2=∏rs∈EL2dr×dsa.
By applying edge partition of Table2,(64)MZ2aL2=3aE1L2×5aE2L2×6aE3L2×4aE4L2×10aE5L2×12aE6L2×6aE7L2×9aE8L2×15aE9L2×18aE10L2=3a1×5a1×6an+5×4a5×10a2×12a2n+8×6a6×9a3n+1×15a2×18a27n+7.
We obtain the following by making some calculations:(65)⇒MZ2aL2=23+8a×31+9a×51+3a×n+52n+83n+127n+7.Theorem 10.
Consider the magnesium iodide structureMgI2 of second type; then, the multiplicative sum and product connectivity indices are equal to(66)SCIIL2=11685n+52n+83n+127n+7,PCIIL2=15415n+52n+83n+127n+7.Proof.
LetL2 be the magnesium iodide structure MgI2 of second type. In Table 2, there is an edge partition of L2. From equation (7),(67)SCIIL2=∏rs∈EL21dr+ds.
By applying edge partition of Table2,(68)SCIIL2=14E1L2×16E2L2×17E3L2×14E4L2×17E5L2×18E6L2×15E7L2×16E8L2×18E9L2×19E10L2=141×161×17n+5×145×172×182n+8×156×163n+1×182×1927n+7.
We obtain the following by making some calculations:(69)⇒SCIIL2=11685n+52n+83n+127n+7.
From equation (8),(70)PCIIL2=∏rs∈EL21dr×ds.
By applying edge partition of Table2,(71)PCIIL2=13E1L2×15E2L2×16E3L2×14E4L2×110E5L2×112E6L2×16E7L2×19E7L2×115E9L2×118E10L2=131×151×16n+5×145×1102×1122n+8×166×193n+1×1152×11827n+7.
We obtain the following by making some calculations:(72)⇒PCIIL2=15415n+52n+83n+127n+7.Theorem 11.
Consider the magnesium iodide structureMgI2 of second type; then, the multiplicative atom-bond connectivity index and geometric-arithmetic index are equal to(73)ABCIIL2=8359n+52n+83n+127n+7,GAIIL2=4801549n+52n+83n+127n+7.Proof.
LetL2 be the magnesium iodide structure MgI2 of second type. In Table 2, there is an edge partition of L2. From equation (9),(74)ABCIIL2=∏rs∈EL2dr+ds−2dr×ds.
By applying edge partition of Table2,(75)ABCIIL2=23E1L2×45E2L2×56E3L2×24E4L2×510E5L2×612E6L236E7L2×46E8L2×615E9L2×718E10L2=231×451×56n+5×245×5102×6122n+8366×3n+149×6152×71827n+7.
We obtain the following by making some calculations:(76)⇒ABCIIL2=8359n+52n+83n+127n+7.
From equation (9),(77)GAIIL2=∏rs∈EL22dr×dsdr+ds.
By applying edge partition of Table2,(78)GAIIL2=234E1L2×256E2L2×267E3L2×244E4L2×2107E5L2×2128E6L2×265E7L2×296E8L2×2158E9L2×2189E10L2=2341×2561×267n+5×2445×21072×21282n+82656×2963n+1×21582×218927n+7.
We obtain the following by making some calculations:(79)⇒GAIIL2=4801549n+52n+83n+127n+7.Theorem 12.
Consider the magnesium iodide structureMgI2 of second type; then, the universal multiplicative geometric-arithmetic index is equal to(80)GAaIIL2=23+2a×7−2a×152+a/2n+52n+83n+127n+7.Proof.
LetL2 be the magnesium iodide structure MgI2 of second type. In Table 2, there is an edge partition of L2. From equation (10),(81)GAaIIL2=∏rs∈EL22dr×dsdr+dsa.
By applying edge partition of Table2,(82)GAaIIL2=234aE1L2×256aE2L2×267aE3L2×244aE4L2×2107aE5L2×2128aE6L2×265aE7L2×296aE8L2×2158aE9L2×2189aE10L2=234a1×256a1×267an+5×244a5×2107a2×2128a2n+8265a6×296a3n+1×2158a2×2189a27n+7.
We obtain the following by making some calculations:(83)⇒GAaIIL2=23+2a×7−2a×152+a/2n+52n+83n+127n+7.


## 3. Numerical and Graphical Comparison

In this section, we compute all the indices numerically and present the result in the following tables.(i)
For the comparison ofII2L1, HII1L1, and HII2L1 indices of MgI2 for m=2n+1, where n∈ℕ, in Table 3, indices for various n values have been calculated. One can clearly see that all the indices are increasing in order for increasing values of n. Graphical representations of these topological indices are shown in Figure 3 for different values of n.(ii)
For the comparison ofSCIIL1, PCIIL1, ABCIIL1, and GAIIL1 indices of MgI2 for m=2n+1, where n∈ℕ, numerically, we computed some indices for different values of n. In Table 4, indices for various n values have been calculated. One can clearly see that all the indices are increasing in order for increasing values of n. Graphical representations of these topological indices are shown in Figure 4 for different values of n.(iii)
For the comparison ofII2L2, HII1L2, and HII2L2 indices of MgI2 for m=2n+2, where n∈ℕ, in Table 5, indices for various n values have been calculated. One can clearly see that all the indices are increasing in order for increasing values of n. Graphical representations of these topological indices are shown in Figure 5 for different values of n.(iv)
For the comparison ofSCIIL2, PCIIL2, ABCIIL2, and GAIIL2 indices of MgI2 for m=2n+2, where n∈ℕ, numerically, we computed some indices for different values of n. In Table 6, indices for various n values have been calculated. One can clearly see that all the indices are increasing in order for increasing values of n. Graphical representations of these topological indices are shown in Figure 6 for different values of n.Figure 3
Lewis structure forMgI2.Figure 4
Graphical comparison amongII2L1, HII1L1, and HII2L1 indices of MgI2 for m=2n+1.Figure 5
Graphical comparison ofSCIIL1, PCIIL1, ABCIIL1, and GAIIL1 indices of MgI2 for m=2n+1.Figure 6
Graphical comparison ofII2L2, HII1L2, and HII2L2 indices of MgI2 for m=2n+2.Table 3
Comparison amongII2L1, HII1L1, and HII2L1 indices of MgI2 for m=2n+1, where n∈ℕ.
nII2L1HII1L1HII2L120.0960×10180.0414×10260.1160×102830.3183×10180.1331×10260.3849×102840.7623×10180.3189×10260.9219×102851.5297×10180.6399×10261.8498×102862.7400×10181.1461×10263.3136×102874.5312×10181.8954×10265.4796×102887.0589×10182.9528×10268.5365×1028910.4970×10184.3910×102612.6943×10281015.0376×10186.2903×102618.1853×1028Table 4
Comparison amongSCIIL1, PCIIL1, ABCIIL1, and GAIIL1 indices of MgI2 for m=2n+1, where n∈ℕ.
nSCIIL1PCIIL1ABCIIL1GAIIL122.9754×1020.0721×1030.2173×1060.2512×10739.8694×1020.2393×1030.7209×1060.8335×107423.6368×1020.5732×1031.7265×1061.9960×107547.4291×1021.1502×1033.4645×1064.0053×107684.9578×1022.0603×1036.2057×1067.1745×1077140.4940×1023.4071×10310.2624×10611.8645×1078218.8690×1025.3079×10315.9872×10618.4831×1079325.4730×1027.8932×10323.7741×10627.4857×10710466.2570×10211.3074×10334.0577×10639.3747×107Table 5
Comparison amongII2L2, HII1L2, and HII2L2 indices of MgI2 for m=2n+2, where n∈ℕ.
nII2L2HII1L2HII2L220.2711×10160.2849×10230.1707×102530.7449×10160.7815×10230.4692×102541.6271×10161.7069×10231.0249×102553.0910×10163.2425×10231.9469×102565.3393×10165.6010×10233.3630×102578.6041×10169.0258×10235.4193×1025813.1469×101613.7912×10238.2806×1025919.2585×101620.2024×102312.1301×10251027.2592×101628.5953×102317.1694×1025Table 6
Comparison amongSCIIL2, PCIIL2, ABCIIL2, and GAIIL2 indices of MgI2 for m=2n+2, where n∈ℕ.
nSCIIL2PCIIL2ABCIIL2GAIIL220.4774×1030.1715×1030.1886×1060.1360×10731.3118×1030.4713×1030.5183×1060.3739×10742.8653×1031.0293×1031.1321×1060.8167×10755.4432×1031.9554×1032.1506×1061.5515×10769.4024×1033.3777×1033.7148×1062.6801×107715.1516×1035.4430×1035.9869×1064.3189×107823.1513×1038.3168×1039.1470×1066.5991×107933.9137×10312.1832×10313.3993×1069.6669×1071048.0028×10317.2445×10318.9658×10613.6830×107


## 4. Discussion

Since topological indices play a vital role in various fields of science such as software engineering, medication, and pharmaceutical industry, their numerical values and graphical representations are very much important for researchers. Here, we calculate some exact values of multiplicative degree-based indices of a famous crystalline structure, magnesium iodideMgI2. Furthermore, we construct Tables 3–6 and Figures 4–7 to estimate the degree-based topological indices for various values of n. From tables and figures, we can see that as n increases, the degree-based indices of these networks also increase.Figure 7
Graphical Comparison amongSCIIL2, PCIIL2, ABCIIL2, and GAIIL2 indices of MgI2 for m=2n+2.


## 5. Conclusion

In this paper, we have constructed the crystalline graph of magnesium iodide with different approaches of graph theory. We subdivided the graph in even and odd vertices. We computed the multiplicative topological indices such as multiplicative Zagreb indices, multiplicative hyper-Zagreb indices, multiplicative universal Zagreb indices, sum and product connectivity of multiplicative indices, multiplicative atom-bond connectivity index, and multiplicative geometric-arithmetic index ofMgI2 structures in this research for the very first time. In addition, we have given general formulation for those indices that may be very beneficial while analyzing the underlying topologies.In future, we are interested to design some new architectures/networks and then study their topological indices, which will be quite helpful to understand their underlying topologies.



---

*Source: 6466585-2022-05-14.xml*