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formerly Central European Journal of Physics

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Volume 16, Issue 1

Issues

Volume 13 (2015)

M-polynomials and topological indices of hex-derived networks

Shin Min Kang
  • Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 52828, Korea
  • Center for General Education, China Medical University, Taichung 40402, Taiwan
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/ Waqas Nazeer / Manzoor Ahmad Zahid / Abdul Rauf Nizami / Adnan Aslam / Mobeen Munir
Published Online: 2018-07-17 | DOI: https://doi.org/10.1515/phys-2018-0054

Abstract

Hex-derived network has a variety of useful applications in pharmacy, electronics, and networking. In this paper, we give general form of the M-polynomial of the hex-derived networksHDN1[n] and HDN2[n], which came out of n-dimensional hexagonal mesh. We also give closed forms of several degree-based topological indices associated to these networks.

Keywords: M-polynomial; topological index; hex-derived network

PACS: 81.05.-t; 81.07.Nb

1 Introduction

In the field of mathematical chemistry, several useful structural and chemical properties of a chemical compound can be determined using simple mathematical tools of polynomials (as Hosoya polynomial and M-polynomial) and numbers (as Randic index and Zagreb index) instead of complicated techniques of quantum mechanics; for details, see Ref. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

The hexagonal mesh was introduced by Chen et al. in 1990 [13]. The 2-dimensional hexagonal mesh HX(2), which is composed of six triangles, is given in Figure 1:

2-dimensional hexagonal mesh
Figure 1

2-dimensional hexagonal mesh

If we add a layer of triangles around this 2-dimensional mesh, we obtain the 3-dimensional hexagonal mesh HX(3); see Figure 2:

3-dimensional hexagonal mesh
Figure 2

3-dimensional hexagonal mesh

The n-dimensional hexagonal mesh HX(n) is obtained by attaching n-2 layers of triangles around HX(2).

A connected planar graph G divides the plane into disjoint regions; each region is called the face of G; two faces are said to be adjacent if they have a common edge; the unbounded region lying outside the graph forms the outer face. The 2-dimensional mesh HX(2) has seven faces, as in Figure 3:

Faces of 2-dimansional hexagonal mesh
Figure 3

Faces of 2-dimansional hexagonal mesh

If corresponding to every closed face f of HX[n] we mark a vertex F and then join it with all the vertices of the face f through edges we receive the hex-derived network HDN1[n]; you can see HDN1 [4] in Figure 4:

The graph of HDN1[4]
Figure 4

The graph of HDN1[4]

If in HDN1[n] the faces f1, f2,…,fk are adjacent to a face f and if the vertex F that represents the face f is joined to the vertices F1, F2, …Fk representing the faces f1, f2, …, fk through edges, we receive the hex-derived network HDN2[n]; one may have a look at HDN2 [4] in Figure 5:

The graph of HDN2[4]
Figure 5

The graph of HDN2[4]

In this report, we give the closed forms of the M-polynomial of HDN1[n] and HDN2[n] and recovered several degree-based topological indices from these polynomials.

Throughout this paper, G will represent a connected graph, V its vertex set, E its edge set, and dv the degree of its vertex v.

Definition 1

The M-polynomial of G is defined as

MG,x,y=δijΔmijGxiyj,(1)

where δ = Min{dv|v ∈ V(G)}, Δ = Max{dv|v ∈ V(G)}, and mij(G) is the edge vuE(G) such that {dv, du} = {i, j}.

Topological indices are graph invariants and presently play an important role in the field of mathematical chemistry, biology, physics, electronics and other applied areas. So for many useful topological indices have been introduced. In 1975, Milan Randic´ introduced the Randic´ index [18], which is defined as

R1/2(G)=uvE(G)1dudv.(2)

For a reasonable information about the development and applications of the Randić index, we refer to [24, 25, 26, 27, 28, 29, 30, 31].

In 1998, working independently, Bollobas and Erdos [19] and Amic et al. [20] proposed the generalized Randic´ index; see [21, 22, 23] for more information.

The general Randic´ index is defined as

Rα(G)=uvE(G)(dudv)α,(3)

and the inverse Randic´ index is defined as RRα(G) = uvE(G)1(dudv)α.

Gutman and Trinajstic´ introduced first Zagreb index and second Zagreb index, which are respectively M1(G) = uvE(G) (du + dv) and M2(G) = uvE(G) (du × dv).

The second modified Zagreb index is defined as

mM2G=uvE(G)1d(u)d(v).(4)

For detailed information, we refer the reader to [32, 33, 34, 35, 36].

The symmetric division index is

SDDG=uvE(G)min(du,dv)max(du,dv)+max(du,dv)min(du,dv).(5)

Another variant of Randic index is the harmonic index, which is defined as

H(G)=vuE(G)2du+dv.(6)

The inverse-sum index is

I(G)=vuE(G)dudvdu+dv.(7)

The augmented Zagreb index is

A(G)=vuE(G)dudvdu+dv23,(8)

which is found useful for computing the heat of formation of alkanes [37, 38].

Some well-known degree-based topological indices are closely related to the M-polynomial [7]; in Table 1 you can see such relations.

Table 1

Derivation of some degree-based topological indices from M-polynomial

Here

Dx=x(fx,yx,Dy=y(fx,yy,Sx=0xft,ytdt,Sy=0yfx,ttdtJfx,y=fx,x,Qαfx,y=xαfx,y.(9)

2 Main results

This section contains the general closed forms of the M polynomial and related indices of the hex-derived networks HDN1[n] and HDN2[n].

Theorem 1

The M-polynomial of HDN1[n], n > 3, is

MHDN1[n];x,y=12x3y5+(18n36)x3y7+(18n254n+42)x3y12+12nx5y7+6x5y12+(6n18)x7y7+(12n24)x7y12+(9n233n+30)x12y12.(10)

Similarly, the edge set E of HDN1[n] can be divided into eight disjoint subsets:

E1(HDN1[n])=e=uvE(HDN1[n]):du=3,dv=5,E2(HDN1[n])=e=uvE(HDN1[n]):du=3,dv=7,E3(HDN1[n])=e=uvE(HDN1[n]):du=3,dv=12,E4(HDN1[n])=e=uvE(HDN1[n]):du=5,dv=7,E5(HDN1[n])=e=uvE(HDN1[n]):du=5,dv=12,E6(HDN1[n])=e=uvE(HDN1[n]):du=7,dv=7,E7(HDN1[n])=e=uvE(HDN1[n]):du=7,dv=12,E8(HDN1[n])=e=uvE(HDN1[n]):du=dv=12.

Also,

E1(HDN1[n])=12,E2(HDN1[n])=18n36,E3(HDN1[n])=18n254n+42,E4(HDN1[n])=12,E5(HDN1[n])=6,

E6(HDN1[n])=6n18,(11)

E7(HDN1[n])=12n24,(12)

E8(HDN1[n])=9n233n+30.(13)

Now, we have

MHDN1[n];x,y=ijmi,jxiyj=35m3,5x3y5+37m3,7x3y7+312m3,12x3y12+57m5,7x5y7+512m5,12x5y12+77m7,7x7y7+712m7,12x7y12+1212m12,12x12y12=uvE1(HDN1[n])m3,5x3y5+uvE2(HDN1[n])m3,7x3y7+uvE3(HDN1[n])m3,12x3y12+uvE4(HDN1[n])m5,7x5y7+uvE5(HDN1[n])m5,12x5y12+uvE6(HDN1[n])m7,7x7y7+uvE7(HDN1[n])m7,12x7y12+uvE8(HDN1[n])1212m12,12x12y12=E1(HDN1[n])x3y5+E2(HDN1[n])x3y7+E3(HDN3[n])x3y12+E4(HDN1[n])x5y7+E5(HDN1[n])x5y12+E6(HDN1[n])x7y7+E7(HDN1[n])x7y12+E8(HDN1[n])x12y12=12x3y5+(18n36)x3y7+(18n254n+42)x3y12+12nx5y7+6x5y12+(6n18)x7y7+(12n24)x7y12+(9n233n+30)x12y12.(35)

Some degree-based topologcal indices of HDN1[n] are given in the following proposition.

Then

Dxf(x,y)=36x3y5+3(18n36)x3y7+3(18n254n+42)x3y12+60nx5y7+30x5y12+7(6n18)x7y7+7(12n24)x7y12+12(9n233n+30)x12y12,

Dyf(x,y)=60x3y5+7(18n36)x3y7+12(18n254n+42)x3y12+84nx5y7+72x5y12+7(6n18)x7y7+12(12n24)x7y12+12(9n233n+30)x12y12,

DyDxf(x,y)=180x3y5+21(18n36)x3y7+36(18n254n+42)x3y12+420nx5y7+360x5y12+49(6n18)x7y7+84(12n24)x7y12+144(9n233n+30)x12y12,

Sy(f(x,y))=125x3y5+17(18n36)x3y7+112(18n254n+42)x3y12+127nx5y7+12x5y12+17(6n18)x7y7+112(12n24)x7y12+112(9n233n+30)x12y12,

SxSy(f(x,y))=1215x3y5+121(18n36)x3y7+136(18n254n+42)x3y12+1235nx5y7+110x5y12+149(6n18)x7y7+184(12n24)x7y12+1144(9n233n+30)x12y12,

DxαDyα(f(x,y))=12×3α5αx3y5+3α7α(18n36)x3y7+3α12α(18n254n+42)x3y12+12×5α7αnx5y7+6×5α12αx5y12+72α(6n18)x7y7+7α12α(12n24)x7y12+122α(9n233n+30)x12y12,

SyDxf(x,y)=365x3y5+37(18n36)x3y7+14(18n254n+42)x3y12+607nx5y7+3012x5y12+(6n18)x7y7+712(12n24)x7y12+(9n233n+30)x12y12,

SxDyf(x,y)=20x3y5+73(18n36)x3y7+4(18n254n+42)x3y12+845nx5y7+725x5y12+(6n18)x7y7+127(12n24)x7y12+(9n233n+30)x12y12,

SxJf(x,y)=32x8+110(18n36)x10+115(18n254n+42)x15+nx12+617x17+114(6n18)x14+119(12n24)x19+124(9n233n+30)x24,

SxJDxDyf(x,y)=452x8+2110(18n36)x10+3615(18n254n+42)x15+35nx12+36017x17+4914(6n18)x14+8419(12n24)x19+6(9n233n+30)x24,

Sx3Q2JDx3Dy3f(x,y)=12×335363x6+337383(18n36)x8+33123133(18n254n+42)x13+12×5373103nx10+6×53123153x15+76(6n18)123x12+73123173(12n24)x17+126223(9n233n+30)x22,

Now, we go for indices.

  1. M1(HDN1[n]) = Dx + Dy(f(x, y))|x=y=1 = 486n2 − 966n + 480

  2. M2(HDN1[n]) = DxDy(f(x, y))|x=y=1 = 1944n2 − 4596n + 2718.

  3. mM2(HDN1[n])=SxSy(f(x,y))x=y=1=1811960310311760n+916n2.

  4. RαHDN1[n]=DxαDyα(f(x,y))x=y=1=12×15α+(18n36)21α+(18n254n+42)36α+12n35α+6×60α+(6n18)49α+(12n24)84α+(9n233n+30)144α.

  5. RRαHDN1[n]=SxαSyα(f(x,y))x=y=1=1215α+18n3621α+18n254n+4236α+12n35α+660α+6n1849α+12n2484α+9n233n+30144α.

  6. SSD(HDN1[n])=SyDx+SxDyf(x,y)x=y=1=3221351265970n+1892n2.

  7. HHDN1[n]=2SxJf(x,y)|x=1=111214522059315320n+6340n2.

  8. I(HDN1[n])=SxJDxDyf(x,y)x=1=25766132301717195n+4865n2.

  9. A(HDN1[n])=Sx3Q2JDx3Dy3f(x,y)x=1=232495775126272675516785532544197414833679806553009155745024n+73828201682924207n2.

Moreover we divide the edge set of HDN2[n] into the following ten partitions:

E1(HDN2[n])=e=uvE(HDN2[n]):du=dv=5,(56)

E2(HDN2[n])=e=uvE(HDN2[n]):du=5,dv=6,(57)

E3(HDN2[n])=e=uvE(HDN2[n]):du=5,dv=7,(58)

E4(HDN2[n])=e=uvE(HDN2[n]):du=5,dv=12,(59)

E5(HDN2[n])=e=uvE(HDN2[n]):du=6,dv=6,(60)

E6(HDN2[n])=e=uvE(HDN2[n]):du=6,dv=7,(61)

E7(HDN2[n])=e=uvE(HDN2[n]):du=6,dv=12,(62)

E8(HDN2[n])=e=uvE(HDN2[n]):du=7,dv=7,(63)

E9(HDN2[n])=e=uvE(HDN2[n]):du=7,dv=12,(64)

and

E10(HDN2[n])=e=uvE(HDN2[n]):du=12,dv=12,(65)

Also, we have

E1(HDN2[n])=18(66)

E2(HDN2[n])=12n24,(67)

E3(HDN2[n])=12n12,(68)

E4(HDN2[n])=6n,(69)

E5(HDN2[n])=9n233n+30,(70)

E6(HDN2[n])=6n12,(71)

E7(HDN2[n])=18n260n+48,(72)

E8(HDN2[n])=6n18,(73)

E9(HDN2[n])=12n24,(74)

and

E10(HDN2[n])=9n233n+30.(75)

Now, by the definition of the M-polynomial, we have

M(HDN2[n];x,y)=ijmi,jxiyj=55m5,5x5y5+56m5,6x5y6+57m5,7x5y7+512m5,12x5y12+66m6,6x6y6+67m6,7x6y7+612m6,12x6y12+77m7,7x7y7+712m7,12x7y12+1212m12,12x12y12=uvE1(HDN2[n])m5,5x5y5+uvE2(HDN2[n])m5,6x5y6+uvE3(HDN2[n])m5,7x5y7+uvE4(HDN2[n])m5,12x5y12+uvE5(HDN2[n])m6,6x6y6+uvE6(HDN2[n])m6,7x6y7+uvE7(HDN2[n])m6,12x6y12+uvE8(HDN2[n])m7,7x7y7+uvE9(HDN2[n])m7,12x7y12+uvE10(HDN2[n])m12,12x12y12=E1(HDN2[n])x5y5+E2(HDN2[n])x5y6+E3(HDN2[n])x5y7+E4(HDN2[n])x5y12+E5(HDN2[n])x6y6+E6(HDN2[n])x6y7+E7(HDN2[n])x6y12+E8(HDN2[n])x7y7+E9(HDN2[n])x7y12+E10(HDN2[n])x12y12=18x5y5+(12n24)x5y6+(12n12)x5y7+6nx5y12+(9n233n+30)x6y6+(6n12)x6y7+(18n260n+48)x6y12+(6n18)x7y7+(12n24)x7y12+(9n233n+30)x12y12.(76)

Now we compute some degree-based topological indices of the hexagonal mesh.

Then

Dyf(x,y)=90x5y5+6(12n24)x5y6+7(12n12)x5y7+72nx5y12+6(9n233n+30)x6y6+7(6n12)x6y7+12(18n260n+48)x6y12+7(6n18)x7y7+12(12n24)x7y12+12(9n233n+30)x12y12,(78)

Dyf(x,y)=90x5y5+6(12n24)x5y6+7(12n12)x5y7+72nx5y12+6(9n233n+30)x6y6+7(6n12)x6y7+12(18n260n+48)x6y12+7(6n18)x7y7+12(12n24)x7y12+12(9n233n+30)x12y12,(79)

DyDx(f(x,y))=450x5y5+(12n24)x5y6+35(12n12)x5y7+360nx5y12+36(9n233n+30)x6y6+42(6n12)x6y7+72(18n260n+48)x6y12+49(6n18)x7y7+84(12n24)x7y12+144(9n233n+30)x12y12,(80)

SxSy(f(x,y))=1825x5y5+130(12n24)x5y6+135(12n12)x5y7+110nx5y12+136(9n233n+30)x6y6+142(6n12)x6y7+172(18n260n+48)x6y12+149(6n18)x7y7+184(12n24)x7y12+1144(9n233n+30)x12y12,(81)

DxαDyα(f(x,y))=18×52αx5y5+5α6α(12n24)x5y6+5α7α(12n12)x5y7+6×5α12αnx5y12+62α(9n233n+30)x6y6+6α7α(6n12)x6y7+6α12α(18n260n+48)x6y12+72α(6n18)x7y7+7α12α(12n24)x7y12+122α(9n233n+30)x12y12,(82)

SxαSyα(f(x,y))=1852αx5y5+15α6α(12n24)x5y6+15α7α(12n12)x5y7+65α12αnx5y12+162α(9n233n+30)x6y6+16α7α(6n12)x6y7+16α12α(18n260n+48)x6y12+172α(6n18)x7y7+17α12α(12n24)x7y12+1122α(9n233n+30)x12y12,(83)

SyDx(f(x,y))=18x5y5+56(12n24)x5y6+57(12n12)x5y7+52nx5y12+(9n233n+30)x6y6+67(6n12)x6y7+12(18n260n+48)x6y12+(6n18)x7y7+712(12n24)x7y12+(9n233n+30)x12y12,(84)

SxDy(f(x,y))=18x5y5+6(12n24)x5y6+75(12n12)x5y7+725nx5y12+(9n233n+30)x6y6+76(6n12)x6y7+2(18n260n+48)x6y12+(6n18)x7y7+127(12n24)x7y12+(9n233n+30)x12y12,(85)

SxJf(x,y)=95x10+111(12n24)x11+(n1)x12+617nx17+112(9n233n+30)x12+113(6n12)x13+118(18n260n+48)x18+114(6n18)x14+119(12n24)x19+124(9n233n+30)x24,(86)

SxJDxDyf(x,y)=45x10+3011(12n24)x11+3512(12n12)x12+36017nx17+3(9n233n+30)x124213(6n12)x13+4(18n260n+48)x18+4914(6n18)x14+8419(12n24)x19+6(9n233n+30)x24,(87)

Sx3Q2JDx3Dy3f(x,y)=94×56x8+536393(12n24)x9+5373103(12n12)x10+6×53123153nx15+66103(9n233n+30)x10+6373113(6n12)x11+63123163(18n260n+48)x16+76123(6n18)x12+73123173(12n24)x17+126223(9n233n+30)x22.(88)

  1. M1(HDN2[n])=Dx+Dy(f(x,y))|x=y=1=648n21500n+852.

  2. M2(HDN2[n])=DxDy(f(x,y))|x=y=1=2916n27566n+4764.

  3. mM2(HDN2[n])=SxSy(f(x,y))|x=y=1=1019329400856311760n+916n2.

  4. Rα(HDN2[n])=DxαDyα(f(x,y))|x=y=1=18×25α+(12n24)30α+(12n12)35α+6n60α+(9n233n+30)36α+(6n12)42α+(18n260n+48)72α+(6n18)49α+(12n24)84α+(9n233n+30)144α.

  5. RRα(HDN2[n])=SxαSyα(f(x,y))|x=y=1=1825α+12n2430α+12n1235α+6n60α+9n233n+3036α+6n1242α+18n260n+4872α+6n1849α+12n2484α+9n233n+30144α.

  6. SSD(HDN2[n])=(SyDx+SxDy)(f(x,y))|x=y=1=1145370n+4325+81n2.

  7. H(HDN2[n])=2SxJ(f(x,y))|x=1=17834871141140271032177759752n+178n2.

  8. I(HDN2[n])=SxJDxDy(f(x,y))x=1=53978927171638134546189n+153n2.

  9. A(HDN2[n])=Sx3Q2JDx3Dy3(f(x,y))|x=1=321800859988717937665809280099878136470440378470436000n+3050648487665500n2.

3 Conclusion

In this article, we computed the M-polynomial of and HDN2(n). The First and the second Zagreb indices, Generalized Randic index, Inverse Randic index, Symmetric division index, Inverse sum index and Augmented Zagreb index of these hex-derived networks have also been computed. These indices are actually functions of chemical graphs and encode many chemical properties as viscosity, strain energy, and heat of formation. Graphical description, given in Figures 6 and 7, also demonstrates the behavior of the M-polynomial of the networks. It is notable that our results about Randic index extend the results given in [39].

The plot for the M-polynomial of HDN1[1]
Figure 6

The plot for the M-polynomial of HDN1[1]

The plot for the M-polynomial of HDN2[1]
Figure 7

The plot for the M-polynomial of HDN2[1]

Acknowledgement

This research is supported by Higher Education Commission of Pakistan.

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About the article

Tel.: +923214707379


Received: 2017-07-10

Accepted: 2018-04-27

Published Online: 2018-07-17


Autor Contributions: All authors contributed equally in writing this paper.

Conflict of InterestConflict of Interests: The authors declare no conflict of interest.


Citation Information: Open Physics, Volume 16, Issue 1, Pages 394–403, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0054.

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© 2018 S. M. Kang et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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