# M-Polynomials and Degree-Based Topological Indices of Triangular, Hourglass, and Jagged-Rectangle Benzenoid Systems

Authors: Young Chel Kwun; Ashaq Ali; Waqas Nazeer; Maqbool Ahmad Chaudhary; Shin Min Kang
Journal: Journal of Chemistry
 (2018)

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## Abstract

Chemical graph theory is a branch of mathematical chemistry which has an important effect on the development of the chemical sciences. The study of topological indices is currently one of the most active research fields in chemical graph theory. Topological indices help to predict many chemical and biological properties of chemical structures under study. The aim of this report is to study the molecular topology of some benzenoid systems.M-polynomial has wealth of information about the degree-based topological indices. We compute M-polynomials for triangular, hourglass, and jagged-rectangle benzenoid systems, and from these M-polynomials, we recover nine degree-based topological indices. Our results play a vital role in pharmacy, drug design, and many other applied areas.


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## Body

## 1. Introduction

Mathematical chemistry provides tools such as polynomials and functions to capture information hidden in the symmetry of molecular graphs and thus predict properties of compounds without using quantum mechanics. A topological index is a numerical parameter of a graph and depicts its topology. Topological indices describe the structure of molecules numerically and are used in the development of qualitative structure activity relationships (QSARs). Most commonly known invariants of such kinds are degree-based topological indices. These are actually the numerical values that correlate the structure with various physical properties, chemical reactivity, and biological activities [1–5]. It is an established fact that many properties such as heat of formation, boiling point, strain energy, rigidity, and fracture toughness of a molecule are strongly connected to its graphical structure.Hosoya polynomial, Wiener polynomial [6], plays a pivotal role in distance-based topological indices. A long list of distance-based indices can be easily evaluated from Hosoya polynomial. A similar breakthrough was obtained recently by Deutsch and Klavžar [7], in the context of degree-based indices. Deutsch and Klavžar [7] introduced M-polynomial in, 2015, to play a role, parallel to Hosoya polynomial to determine closed form of many degree-based topological indices [8–11]. The real power of M-polynomial is its comprehensive nature containing healthy information about degree-based graph invariants. These invariants are calculated on the basis of symmetries present in the 2d-molecular lattices and collectively determine some properties of the material under observation. Benzenoid hydrocarbons play a vital role in our environment and in the food and chemical industries.Benzenoid molecular graphs are systems with deleted hydrogens. It is a connected geometric figure obtained by arranging congruent regular hexagons in a plane so that two hexagons are either disjoint or have a common edge.This figure divides the plane into one infinite (external) region and a number of finite (internal) regions. All internal regions must be regular hexagons. Benzenoid systems are of considerable importance in theoretical chemistry because they are the natural graph representation of benzenoid hydrocarbons. A vertex of a hexagonal system belongs to, at most, three hexagons. A vertex shared by three hexagons is called an internal vertex [12].In this paper, we study three benzenoid systems, namely, triangular, hourglass, and jagged-rectangle benzenoid systems.


## 2. Basic Definitions and Literature Review

Throughout this article, we assumeG to be a connected graph, V (G) and E (G) are the vertex set and the edge set, respectively, and dv denotes the degree of a vertex v.Definition 1. The M-polynomial of G is defined as MG,x,y=∑δ≤i≤j≤ΔmijGxiyj where δ=Mindv∣v∈VG,Δ=Maxdv∣v∈VG and mijG is the edge vu∈EG such that where ≤j [7].Wiener Index and its various applications are discussed in [13–15]. Randić Index, R−1/2G, was introduced by Milan Randić in 1975, defined as R−1/2G=∑uv∈EG1/dudv. For general details about R−1/2G and its generalized Randić Index, RαG=∑uv∈EG1/dudvα, refer [16–20], and the Inverse Randić Index is defined as RRαG=∑uv∈EGdudvα. Clearly, R−1/2G is a special case of RαG where α=−1/2. This index has many applications in diverse areas. Many papers and books such as [21–23] are written on this topological index as well. Gutman and Trinajstić introduced two indices defined as M1G=∑uv∈EGdudv and M2G=∑uv∈EGdudv. The modified second Zagreb Index is defined as mM2G=∑uv∈EG1/dudv. We refer [24–28] to the readers for comprehensive details of these indices. Other famous indices are Symmetric Division Index: SDD(G) = ∑uv∈EGmindu,dv/maxdu,dv+maxdu,dv/mindu,dv, Harmonic Index: HG=∑vu∈EG2/du+dv, Inverse Sum Index: IG=∑vu∈EGdudv/du+dv, and Augmented Zagreb Index: AG=∑vu∈EGdudv/du+dv−23 [29, 30].Tables presented in [7–10] relate some of these well-known degree-based topological indices with M-polynomial with the following reserved notations:(1)Dx=x∂fx,y∂x,Dy=y∂fx,y∂y,Sx=∫0xft,ytdt,Sy=∫0yfx,ttdt,Jfx,y=fx,x,Qαfx,y=xαfx,y.


## 3. Computational Results

In this section, we give our computational results. In terms of chemical graph theory and mathematical chemistry, we associate a graph with the molecular structure where vertices correspond to atoms and edges to bonds. The triangular benzenoid system is shown in Figure1. In the following theorem, we compute M-polynomial of the triangular benzenoid system.Figure 1
Triangular benzenoid.Theorem 1.
LetTp be a Triangular benzenoid system where p shows the number of hexagons in the base graph and total no. of hexagons in Tp is 1/2pp+1. Then,(2)MTp,x,y=6x2y2+6p−1x2y3+32pp−1x3y3.Proof.
LetTp be a triangular benzenoid. Then from Figure 1, we have(3)VTp=p2+4p+1,ETp=32pp+3.The edge set ofTp has the following three partitions:(4)E2,2=e=uv∈ETpdTpu=2,dTpv=2,E2,3=e=uv∈ETpdTpu=2,dTpv=3,E3,3=e=uv∈ETpdTpu=3,dTpv=3.Now,(5)E2,2=6,E2,3=6p−1,E3,3=32pp−1.Thus, theM-polynomial of Tp is(6)MTp;x,y=∑i≤jmijTpxiyj=∑2≤2m22Tpx2y2+∑2≤3m23Tpx2y3+∑3≤3m33Tpx3y3=∑uv∈E2,2m22Tpx2y2+∑uv∈E2,3m23Tpx2y3+∑uv∈E3,3m33Tpx3y3=E2,2x2y2+E2,3x2y3+E3,3x3y3=6x2y2+6p−1x2y3+32pp−1x3y3.Now, we derive formulas for many degree-based topological indices usingM-polynomial.Proposition 2.
LetTp be a triangular Benzenoid.Then,(1)
M1Tp=9p2+21p−6(2)
M2Tp=27/2p2+45p−12(3)
Mm2Tp=1/6p2+5/6p+1/2(4)
RαTp=32α+1/2p2+2α+1⋅3α+1−32α+1/2p+3⋅22α+1−2α+1⋅3α+1(5)
RRαTp=1/2⋅32α−1p2+1/2α−1⋅3α−1+1/2⋅32α−1p+3/22α−1−1/2α−1⋅3α−1(6)
SSDTp=3p2+24p−9(7)
HTp=1/2p2+19/10p+3/5(8)
ITp=9/4p2+99/20p−6/5(9)
ATp=2187/128p2+46965/128p−336Proof.
Let(7)MTp,x,y=6x2y2+6p−1x2y3+32pp−1x3y3.
Then,(8)Dxfx,y=12x2y2+12p−1x2y3+92pp−1x3y3,Dyfx,y=12x2y2+18p−1x2y3+92pp−1x3y3,DyDxfx,y=24x2y2+36p−1x2y3+272pp−1x3y3,SxSyfx,y=32x2y2+p−1x2y3+16pp−1x3y3,DxαDyαfx,y=3⋅22α+1x2y2+2α+1⋅3α+1p−1x2y3+32α+12pp−1x3y3,SxαSyαfx,y=12⋅32α−1p2+12α−1⋅3α−1−12⋅32α−1p+322α−1−12α−1⋅3α−1,SyDxfx,y=6x2y2+4p−1x2y3+32pp−1x3y3,SxDyfx,y=6x2y2+9p−1x2y3+32pp−1x3y3,SxJfx,y=32x4+65p−1x5+14pp−1x6,SxJDxDyfx,y=6x4+365p−1x5+94pp−1x6,Sx3Q−2JDx3Dy3fx,y=48x2+384p−1x3+2187128pp−1x4.(1)
First Zagreb Index:(9)M1Tp=Dx+Dyfx,yx=y=1=9p2+21p−6.(2)
Second Zagreb Index:(10)M2Tp=DyDxfx,yx=y=1=272p2+45p−12.(3)
Modified second Zagreb Index:(11)Mm2Tp=SxSyfx,yx=y=1=16p2+56p+12.(4)
Generalized Randic Index:(12)RαTp=DxαDyαfx,yx=y=1=32α+12p2+2α+1⋅3α+1−32α+12p+3⋅22α+1−2α+1⋅3α+1.(5)
Inverse Randic Index:(13)RRαTp=SxαSyαfx,yx=y=1=12⋅32α−1p2+12α−1⋅3α−1+12⋅32α−1p+322α−1−12α−1⋅3α−1.(6)
Symmetric Division Index:(14)SSDTp=SyDx+SxDyfx,yx=y=1=3p2+24p−9.(7)
Harmonic Index:(15)HTp=2SxJfx,yx=1=12p2+1910p+35.(8)
Inverse Sum Index:(16)ITp=SxJDxDyfx,yx=1=94p2+9920p−65.(9)
Augmented Zagreb Index:(17)ATp=Sx3Q−2JDx3Dy3fx,yx=1=2187128p2+46965128p−336.
Topological indices ofTp for specific values of p are given in Table 1.
Our next target is the benzenoid hourglass system which is obtained from two copies of a triangular benzenoidTp by overlapping their external hexagons and shown in Figure 2. In Theorem 3, we compute M-polynomial of the benzenoid hourglass system.Table 1
Topological indices of triangular benzenoid for different values ofp.
Name of index
p = 1
p = 2
p = 3
p = 4
p = 5
M-polynomial
6
x
2
y
2
6
x
2
y
2
+
6
x
2
y
3
+
3
x
3
y
3
6
x
2
y
2
+
12
x
2
y
3
+
9
x
3
y
3
6
x
2
y
2
+
18
x
2
y
3
+
18
x
3
y
3
6
x
2
y
2
+
24
x
2
y
3
+
30
x
3
y
3
First Zagreb Index
24
72
138
222
324
Second Zagreb Index
46.50000000
132
244.5000000
384
550.5000000
Modified second Zagreb Index
1.500000000
2.833333333
4.500000000
6.500000000
8.833333333
Randić Index forα=1/2
17.39387692
41.09081537
73.78775383
115.4846923
166.1816308
Inverse Randić Index forα=1/2
2.898979486
7.348469229
12.79795897
19.24744872
26.69693846
Symmetric Division Index
18
5
90
135
186
Harmonic Index
3
6.400000000
10.80000000
16.20000000
22.60000000
Inverse Sum Index
6
17.70000000
33.90000000
27.30000000
39.90000000
Augmented Zagreb Index
48
466.1718750
918.5156250
1405.031250
1925.718750Figure 2
Benzenoid hourglass system.Theorem 3.
LetXp denotes the Benzenoid Hourglass. Then, its M-polynomial is(18)MXp,x,y=8x2y2+43p−4x2y3+3p2−3p+4x3y3.Proof.
LetXp denotes the benzenoid hourglass which is obtained from two copies of a triangular benzenoid Tp by overlapping their external hexagons. Then, we have(19)VXp=2p2+4p−2,EXp=3p2+9p−4.
The edge set ofXp has the following three partitions:(20)E2,2=e=uv∈EXpdXpu=2,dXpv=2,E2,3=e=uv∈EXpdXpu=2,dXpv=3,E3,3=e=uv∈EXpdXpu=3,dXpv=3.
Now,(21)E2,2=8,E2,3=43p−4,E3,3=3p2−3p+4.
Thus, theM-polynomial of Xp is(22)MXp;x,y=∑i≤jmijXpxiyj=∑2≤2m22Xpx2y2+∑2≤3m23Xpx2y3+∑3≤3m33Xpx3y3=∑uv∈E2,2m22Xpx2y2+∑uv∈E2,3m23Xpx2y3+∑uv∈E3,3m33Xpx3y3=E2,2x2y2+E2,3x2y3+E3,3x3y3=8x2y2+43p−4x2y3+3p2−3p+4x3y3.
Now, we derive formulas for many degree-based topological indices usingM-polynomial.Proposition 4.
LetXp be a Benzenoid Hourglass.Then,(1)
M1Xp=63p2+7p−4(2)
M2Xp=27p2+45p−28(3)
Mm2Xp=1/3p2+5/3p−2/9(4)
RαXp=32α+1p2+2α+2⋅3α+1−32α+1p+22α+3−2α+4⋅3α+4⋅32α(5)
RRαXp=1/32α−1p2+1/3α−1⋅2α−2−1/32α−1p+1/22α−3−1/3⋅α2α−4+4/32α(6)
SSDXp=6p2+20p−32/3(7)
HXp=p2+19/5p−16/15(8)
IXp=9/2p2+99/10p−26/5(9)
AXp=2187/64p2+3957/64p−295/16Topological indices ofXp for specific values of p are given in Table 2.Table 2
Topological indices of benzenoid hourglass for different values ofp.
Name of index
p = 1
p = 2
p = 3
p = 4
p = 5
M-polynomial
8
x
2
y
2
−
4
x
2
y
3
+
4
x
3
y
3
8
x
2
y
2
+
8
x
2
y
3
+
10
x
3
y
3
8
x
2
y
2
+
20
x
2
y
3
+
22
x
3
y
3
8
x
2
y
2
+
32
x
2
y
3
+
40
x
3
y
3
8
x
2
y
2
+
44
x
2
y
3
+
64
x
3
y
3
First Zagreb Index
36
132
264
432
636
Second Zagreb Index
44
170
350
584
872
Modified second Zagreb Index
1.777777778
4.444444444
7.777777778
11.77777778
16.44444444
Randić Index forα=1/2
92.81601318
198.9976439
323.1792746
465.3609054
625.5425361
Inverse Randić Index forα=1/2
3.648129059
8.097618797
14.54710855
22.99659829
33.44608803
Symmetric Division Index
15.33333333
53.33333333
103.3333333
165.3333333
239.3333333
Harmonic Index
3.733333333
10.53333333
19.33333333
30.13333333
44.26666667
Inverse Sum Index
9.200000000
32.60000000
65
106.4000000
156.8000000
Augmented Zagreb Index
77.56250000
241.9062500
474.5937500
775.6250000
1145Now, we study benzenoid jagged-rectangle shown in Figure3.Figure 3
Benzenoid jagged-rectangle system.Theorem 5.
LetBp,q denotes a Jagged-rectangle Benzenoid system for all p,q∈N−1.Then,(23)MBp,q;x,y=2q+2x2y2+4p+q−1x2y3+6pq+p−5q−4x3y3.Proof.
LetBp,q denotes a benzenoid system jagged-rectangle for all p,q∈N−1. A benzenoid jagged-rectangle forms a rectangle and the number of benzenoid called in each chain alternate p and p − 1.
The edge set ofBp,q has the following three partitions:(24)E2,2=e=uv∈EBp,qdBp,qu=2,dBp,qv=2,E2,3=e=uv∈EBp,qdBp,qu=2,dBp,qv=3,E3,3=e=uv∈EBp,qdBp,qu=3,dBp,qv=3.
Now,(25)E2,2=2q+4,E2,3=4p+4q−4,E3,3=6pq+p−5q−4.
Thus, theM-polynomial of Bp,q is(26)MBp,q;x,y=∑i≤jmijBp,qxiyj=∑2≤2m22Bp,qx2y2+∑2≤3m23Bp,qx2y3+∑3≤3m33Bp,qx3y3=∑uv∈E2,2m22Bp,qx2y2+∑uv∈E2,3m23Bp,qx2y3+∑uv∈E3,3m33Bp,qx3y3=E2,2x2y2+E2,3x2y2+E3,3x3y3=2q+4x2y2+4p+q−1x2y3+6pq+p−q−4x3y3=2q+2x2y2+4p+q−1x2y3+6pq+p−q−4x3y3.Proposition 6.
LetBp,q denotes a Jagged-rectangle Benzenoid system for all p,q∈N−1.Then,(1)
M1Bp,q=213p−q+18pq−14(2)
M2Bp,q=33p−13q−44(3)
Mm2Bp,q=7/9p+11/18q+2/3pq−1/9(4)
RαBp,q=3α⋅2α+2+32αp+22α+1+3α⋅2α+2−5⋅32αq+6⋅32αpq+22α+2−3α⋅2α+2−4⋅32α(5)
RRαBp,q=1/3α⋅2α−2+1/32αp+1/22α−1+1/3α⋅2α−2−5/32αq+6/32αpq+1/22α−2−1/3⋅α2α−2−4/32α(6)
SSDBp,q=32/3p+8/3q+12pq−26/3(7)
HBp,q=29/15p+14/15q+2pq−14/15(8)
IBp,q=63/10p−7/10q+9pq−34/5(9)
ABp,q=2777/64p−573/64q+2187/32pq−729/16.Topological indices ofBp,q for specific values of p and q are given in Table 3.Table 3
Topological indices of jagged-rectangle benzenoid system for different values ofp and q.
Name of index
p = 1, q = 1
p = 2, q = 11
p = 3, q = 11
p = 4, q = 11
p = 5, q = 11
M-polynomial
6
x
2
y
2
+
4
x
2
y
3
−
2
x
3
y
3
6
x
2
y
2
+
8
x
2
y
3
+
4
x
3
y
3
6
x
2
y
2
+
12
x
2
y
3
+
12
x
3
y
3
6
x
2
y
2
+
16
x
2
y
3
+
16
x
3
y
3
6
x
2
y
2
+
20
x
2
y
3
+
22
x
3
y
3
First Zagreb Index
12
54
96
138
180
Second Zagreb Index
−24
9
42
75
108
Modified second Zagreb Index
1.944444444
3.388888889
4.833333333
6.277777778
7.722222222
Randic Index forα=1/2
15.79795897
46.59591794
77.39387692
108.1918359
138.9897949
Inverse Randic Index forα=1/2
3.966326495
7.932652990
11.89897949
15.86530598
19.83163248
Symmetric Division Index
16.66666667
39.33333333
62
84.66666667
107.3333333
Harmonic Index
3.933333333
7.866666667
11.80000000
15.73333333
19.66666667
Inverse Sum Index
9.200000000
24.50000000
39.80000000
55.10000000
70.40000000
Augmented Zagreb Index
57.21875000
168.9531250
280.6875000
392.4218750
504.1562500


## 4. Conclusion

In this paper, we computedM-polynomials for triangular, hourglass, and jagged-rectangle benzenoid systems. From these M-polynomials, we recover first Zagreb, second Zagreb, modified second Zagreb, Randić, inverse Randić, symmetric division, inverse sum, and harmonic and augmented Zagreb indices of triangular, hourglass, and jagged-rectangle benzenoid systems. Note that there are no benzenoid molecules having the triangular graphs and hourglass graphs as their skeleton. It is important to mention here that some of these topological indices are calculated directly by using formulas in the literature.



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