Algebraic Properties for Molecular Structure of Magnesium Iodide

Ali N.A.Koam,Ali Ahmad,Muhammad Azeem and Muhammad Kamran Siddiqui

1Department of Mathematics,College of Science,New Campus,Jazan University,Jazan,Saudi Arabia

2College of Computer Science and Information Technology,Jazan University,Jazan,Saudi Arabia

3Department of Mathematics,Riphah Institute of Computing and Applied Sciences,Riphah International University,Lahore,Pakistan

4Department of Mathematics,COMSATS University Islamabad,Lahore,Pakistan

ABSTRACT As an inorganic chemical,magnesium iodide has a significant crystalline structure.It is a complex and multifunctional substance that has the potential to be used in a wide range of medical advancements.Molecular graph theory,on the other hand,provides a sufficient and cost-effective method of investigating chemical structures and networks.M-polynomial is a relatively new method for studying chemical networks and structures in molecular graph theory.It displays numerical descriptors in algebraic form and highlights molecular features in the form of a polynomial function.We present a polynomials display of magnesium iodide structure and calculate several M-polynomials in this paper,particularly the M-polynomials of the augmented Zagreb index,inverse sum index,hyper Zagreb index and for the symmetric division index.

KEYWORDS Magnesium iodide;M-polynomials;algebraic properties of magnesium iodide;algebraic formation of numerics of magnesium iodide

1 Introduction

The chemical formulaMX2is referred to a chemical compound named magnesium iodide.It has numerous commercial uses and is also valuable to get different organic synthesis,noted thatMX2or magnesium iodide is an inorganic compound.Nanopowder,submicron,high impurity and volumes are the availability measures ofMX2.It is a very useful asset in internal medicine.Magnesium iodide may be produced by reacting hydro-iodic acid with magnesium carbonate,magnesium hydroxide,and magnesium oxide.Magnesium iodide is made up of a uniqueC4-graph pattern,according to molecular graph theory.Multiple heptagons are connected to each other,having eachC4-graph inside[1].For a better understanding of the magnesium iodide molecular graph,we labeled the parameters as follows:mis the number ofC4’s of upper sides in a row,andnis the number of lower sideC4 in heptagons.The magnesium iodide graph is required to preserve the relationship ofmfor even and odd values of separately with the relation ofm=2(n+1),andm=2n+1,respectively,for all values ofn∈Z withn≥1.

In this article,we studied some M-polynomials of magnesium iodide structure for both cases ofm.The studied M-polynomials are defined below in Definitions 1.1 to 1.5,with their other fundamentals.Moreover,for a graphχand their polynomial parameters are,?,whileξais the degree of a vertex a and defined by the counting of edges attached to a vertex a.

Definition 1.1.The most famous and first,Hosoya polynomials were introduced by [2] in 1988,and in 2015,Modified-polynomial,also known as M-polynomial,is introduced by[3].This form of the polynomial has a closed relationship to degree-based topological indices.Topological indices from Mpolynomials of a graph may be obtained using a specific format.This M-polynomial may be defined as follows:

wheremi,j(χ)is the number of edges of graphχsuch thati≤j,are the notation of degrees of a vertex.

Definition 1.2.The symmetric division index SDI of a graphχwas introduced by [4] in 2010.Following are the symmetric division index and its M-polynomials.

whereD,D?,S?,Sare differential and integral operators defined in the Eqs.(10)to(15).

Definition 1.3.Shirdel et al.[5,6]introduced hyper Zagreb index,and given below are topological descriptor and its M-polynomial.

Definition 1.4.Inverse sum index introduced by [7],and its M-polynomials [8],given below are formulations of its indices and M-polynomial.

Definition 1.5.Furtula et al.[9]defined augmented Zagreb index and given below are formulation of index and its M-polynomial.

where

The researchers in[3]introduced the concept,in combination with numerical descriptors(known as topological indices[10–13]),an algebraic graph theory.The concept is known as the M-polynomials of the graph.It has fundamentals from topological indices.For some basic and important topological indices,we refer to see the articles[14–17].This subject has been extensively researched in the last halfdecade,and there is a wealth of material accessible.We will review a few of the most current papers on this topic.

The study of M-polynomials on nanotubes are available for h-naphthenic nanotube[18],various nanostructures studied in [19],andVC5C7-type of nanotubes are discussed in [20].Generalized classes and families of the graph are studied with different M-polynomials in [15,21,22],various benzenoid structures are showcased in [23],M-polynomials interaction with statistics probability concept are available in[24].Metal organic chemical network’s M-polynomials are studied in[25,26].M-polynomials concept is also studied for various computer related networks and found in[27,28].

The M-polynomials are figured out for the magnesium iodide orMX2structure for both even and odd cases of parameterm,and such M-polynomials named of augmented Zagreb index,inverse sum index,hyper Zagreb index and the symmetric division index.

2 Results on the M-Polynomials of Magnesium Iodide

Given in this section are some important results of this research work.The idea totally depends on the structural values of theMX2or magnesium iodide graph,which is defined in Table 1(form=odd)and Table 2(forn=even).

Table 1: Edge partition of I(m,n)m=2n+1

Table 1 (continued)(ξa,ξb)FrequencySet of edges(2,5)8E6(2,6)2n+8E7(3,3)3nE8(3,4)1E9(3,5)12E10(3,6)27n?13E11

Case 1:For the odd values ofmwith givenn≥1.Letm=2n+1 andn∈Z.

Theorem 2.1.LetIm,nbe a magnesium iodide graph,withm=2n+1,n≥1,shown in Fig.1.Then,its M-polynomial is

Figure 1:Magnesium iodide graph for n=4 and m=7

Proof.The construction of magnesium iodide graph from its structure,which is found in Fig.1.Ithas six type of vertices,evaluated from the figure and these vertices are described in the vertex set as:

The same figure also shows the information of its edge types.It contains eleven types of edges in total and these are described as:

The cardinality of these edge partitions areThen from the Definition 1.1,the M-polynomial ofIm,nis

The 3D plot shown in the Fig.2,is the general M-polynomial of magnesium iodide graph,for a particular value withn=7 andm=15.

Figure 2:Graphical plot of M-polynomial of magnesium iodide graph

Lemma 2.1.LetIm,nbe a magnesium iodide graph,withm=2n+1,n≥1,Then the differential operators are

Proof.Differentiate Eq.(16) with respect toand multiply the result with,we get theSimilarly,differentiate Eq.(16) with respect to ? and multiply the result with ?,we get the

Lemma 2.2.LetIm,nbe a magnesium iodide graph,withm=2n+1,n≥1.Then the integral operators are

Proof.As we know that from Eq.(12),uses the general Mpolynomial for theIm,nfrom Eq.(16)in it.After simplification,we obtainSimilarly,from Eq.(13),uses the general M-polynomial for theIm,nfrom Eq.(16)in it.After simplification,we obtain

Lemma 2.3.LetIm,nbe a magnesium iodide graph,withm=2n+1,n≥1.Then the Eq.(16)is

Proof.By implementing the operator stated in the Eq.(14) on the main equation defined in Eq.(16),we get the

Theorem 2.2.LetIm,nbe a magnesium iodide graph,withm=2n+ 1,n≥1,andPSDIis the M-polynomial of symmetric division index.Thenis

Proof.The M-polynomial of symmetric division index,given in the Definition 1.2,and the formulas for our structureIm,ncan be written as:PSDI(Im,n)=(M(χ;,?)).Lemmas 2.1 and 2.2 are the operators and here we used these to find our results forIm,n.We will obtain the M-polynomial of symmetric division index forIm,nafter some algebraic simplifications as

The 3D plot shown in the Fig.3,is the symmetric division index M-polynomial of magnesium iodide graph,for a particular value withn=7 andm=15.

Figure 3:Graphical plot of SDI M-polynomial of magnesium iodide graph

Theorem 2.3.LetIm,nbe a magnesium iodide graph,withm=2n+ 1,n≥1,andPHMis the M-polynomial of hyper Zagreb index.Thenis

Proof.The M-polynomial of hyper Zagreb index,given in the Definition 1.3,and the formulas for our structureIm,ncan be written as:PHM(Im,n)=(M(χ;,)).Lemmas 2.2 and 2.3 are the operators and here we used these to find our results forIm,n.We will obtain the M-polynomial of symmetric division index forIm,nafter some algebraic simplifications as:

The 2D plot shown in the Fig.4,is the hyper Zagreb M-polynomial of magnesium iodide graph.Moreover,all the 2D plots show the M-polynomials of different index containingvalues on horizontal scale with different values ofn∈{2,3,...,10}.

Figure 4:Graphical plot of HM M-polynomial of magnesium iodide graph

Theorem 2.4.LetIm,nbe a magnesium iodide graph,withm=2n+ 1,n≥1,andPISis the M-polynomial of inverse sum index.Thenis

Proof.The M-polynomial of inverse sum index,given in the Definition 1.4,and the formulas for our structureIm,ncan be written as:PIS(Im,n)=Lemmas 2.1,2.2 and 2.3 are the operators and here we used these to find our results forIm,n.We will obtain the M-polynomial of symmetric division index forIm,nafter some algebraic simplifications as

The 2D plot shown in the Fig.5,is the inverse sum M-polynomial of magnesium iodide graph,for a particular value withn=7 andm=15.

Figure 5:Graphical plot of IS M-polynomial of magnesium iodide graph

Theorem 2.5.LetIm,nbe a magnesium iodide graph,withm=2n+ 1,n≥1,andPAZIis the M-polynomial of augmented Zagreb index.Thenis

Proof.The M-polynomial of augmented Zagreb index,given in the Definition 1.5,and the formulas for our structureIm,ncan be written as:PAZI(Im,n)=Lemmas 2.1,2.2 and 2.3 are the operators and here we used these to find our results forIm,n.We will obtain the M-polynomial of symmetric division index forIm,nafter some algebraic simplifications as

The 2D plot shown in the Fig.6,is the augmented Zagreb index M-polynomial of magnesium iodide graph,for a particular value withn=7 andm=15.

Case 2:For the even values ofmwith givenn≥1.Letm=2(n+1)andn∈Z.

Theorem 2.6.LetIm,nbe a magnesium iodide graph,withm=2(n+1),n≥1,shown in the Fig.1.Then,its M-polynomial is

Figure 6:Graphical plot of AZI M-polynomial of magnesium iodide graph

Proof.The construction of magnesium iodide graph from its structure is found in Fig.1.It has five types of vertices,evaluated from the figure and these vertices are described in the vertex set as

The same figure is also shown the information of its edge types.It contains ten types of edges in total and these are described as

The cardinality of these edge partitions areand=27n+7.Then from the definition 1.1,the M-polynomial ofIm,nis

Table 2: Edge partition of I(m,n)for m=2(n+1)

The 3D plot shown in the Fig.7,is the general M-polynomial of magnesium iodide graph,for a particular value withn=7 andm=16.

Lemma 2.4.LetIm,nbe a magnesium iodide graph,withm=2(n+1),n≥1,then the differential operators are

Proof.Differentiate Eq.(26) with respect toand multiply the result with,we get theSimilarly,differentiate Eq.(26) with respect to ? and multiply the result with ?,we get the

Figure 7:Graphical plot of M-polynomial of magnesium iodide graph

Lemma 2.5.LetIm,nbe a magnesium iodide graph,withm=2n+1,n≥1.Then the integral operators are

Proof.As we know that from Eq.(12),dtuses the general Mpolynomial for theIm,nfrom Eq.(26)in it.After simplification,we obtainSimilarly,from Eq.(13),dtuses the general M-polynomial for theIm,nfrom Eq.(26)in it.After simplification,we obtain

Lemma 2.6.LetIm,nbe a magnesium iodide graph,withm=2(n+1),n≥1.Then the Eq.(26)is

Proof.By implementing the operator stated in the Eq.(14) on the main equation defined in 11,we get the

Theorem 2.7.LetIm,nbe a magnesium iodide graph,withm=2(n+1),n≥1,andPSDIis the M-polynomial of symmetric division index.Thenis

Proof.The M-polynomial of symmetric division index,given in the Definition 1.2,and the formulas for our structureIm,ncan be written as:PSDI(Im,n)=(M(χ;,?)).Lemmas 2.4 and 2.5 are the operators and here we used these to find our results forIm,n.We will obtain the M-polynomial of symmetric division index forIm,nafter some algebraic simplifications as

The 3D plot shown in the Fig.8,is the symmetric division index M-polynomial of magnesium iodide graph,for a particular value withn=7 andm=16.

Theorem 2.8.LetIm,nbe a magnesium iodide graph,withm=2(n+1),n≥1,andPHMis the M-polynomial of hyper Zagreb index.Thenis

Proof.The M-polynomial of hyper Zagreb index,given in the Definition 1.3,and the formulas for our structureIm,ncan be written as:PHM(Im,n)=2SJ(M(G;,?)).Lemmas 2.5 and 2.6 are the operators and here we used these to find our results forIm,n.We will obtain the M-polynomial of symmetric division index forIm,nafter some algebraic simplifications as

Figure 8:Graphical plot of SDI M-polynomial of magnesium iodide graph

The 2D plot shown in the Fig.9,is the hyper Zagreb M-polynomial of magnesium iodide graph,for a particular value withn=7 andm=16.

Theorem 2.9.LetIm,nbe a magnesium iodide graph,withm=2(n+1),n≥1,andPISis the M-polynomial of inverse index.Thenis

Figure 9:Graphical plot of HM M-polynomial of magnesium iodide graph

Proof.The M-polynomial of inverse sum index,given in the Definition 1.4,and the formulas for our structureIm,ncan be written as:PIS(Im,n)=SJDD?(M(χ;,?)).Lemmas 2.4,2.5 and 2.6 are the operators and here we used these to find our results forIm,n.We will obtain the M-polynomial of symmetric division index forIm,nafter some algebraic simplifications as

The 2D plot shown in the Fig.10,is the inverse sum M-polynomial of magnesium iodide graph,for a particular value withn=7 andm=16.

Figure 10:Graphical plot of IS M-polynomial of magnesium iodide graph

Theorem 2.10.LetIm,nbe a magnesium iodide graph,withm=2(n+1),n≥1,andPAZIis the M-polynomial of augmented Zagreb index.Thenis

Proof.The M-polynomial of augmented Zagreb index,given in the Definition 1.5,and the formulas for our structureIm,ncan be written as:PAZI(Im,n)=Lemmas 2.4,2.5 and 2.6 are the operators and here we used these to find our results forIm,n.We will obtain the M-polynomial of symmetric division index forIm,nafter some algebraic simplifications as

The 2D plot shown in the Fig.11,is the augmented Zagreb index M-polynomial of magnesium iodide graph,for a particular value withn=7 andm=16.

Figure 11:Graphical plot of AZI M-polynomial of magnesium iodide graph

3 Conclusion

The M-polynomial of a structure delivers the polynomial or abstract function of a chemical network or structure.To add up some algebraic properties of magnesium iodide orMX2structure,we computed the M-polynomials of augmented Zagreb index,inverse sum index,hyper Zagreb index and the symmetric division index are figured out for the magnesium iodide orMX2structure,for both even and odd cases of parameterm.

Funding Statement:The authors received no specific funding for this study.

Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.