M-polynomials and topological indices of hex-derived networks
-
Shin Min Kang
,
Manzoor Ahmad Zahid
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.
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
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
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
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:
![Figure 4 The graph of HDN1[4]](/document/doi/10.1515/phys-2018-0054/asset/graphic/j_phys-2018-0054_fig_004.jpg)
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:
![Figure 5 The graph of HDN2[4]](/document/doi/10.1515/phys-2018-0054/asset/graphic/j_phys-2018-0054_fig_005.jpg)
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
where δ = Min{dv|v ∈ V(G)}, Δ = Max{dv|v ∈ V(G)}, and mij(G) is the edge vu ∈ E(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
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
and the inverse Randic´ index is defined as RRα(G) =
Gutman and Trinajstic´ introduced first Zagreb index and second Zagreb index, which are respectively M1(G) =
The second modified Zagreb index is defined as
For detailed information, we refer the reader to [32, 33, 34, 35, 36].
The symmetric division index is
Another variant of Randic index is the harmonic index, which is defined as
The inverse-sum index is
The augmented Zagreb index is
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.
Derivation of some degree-based topological indices from M-polynomial
| Topological Index | Derivation from M(G; x, y) |
|---|---|
| First Zagreb | (Dx + Dy) (M(G; x, y))x=y=1 |
| Second Zagreb | (DxDy)(M(G; x, y))x=y=1 |
| Second Modified | (SxSy) (M(G; x, y))x=y=1 |
| Zagreb | |
| Inverse Randic | |
| General Randic | |
| Symmetric Division | (DxSy + SxDy) (M(G; x, y))x=y=1 |
| Index | |
| Harmonic Index | 2SxJ(M(G; x, y))x=1 |
| Inverse Sum Index | SxJDxDy(M(G;x, y))x=1 |
| Augmented Zagreb | Sx3 Q–2JDx3Dy3(M(G; x, y))x=1 |
| Index |
Here
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
Proof
Depending on degrees, the vertex set E of HDN1[n] can be divided into four disjoint subsets, V1, V2, V3, and V4, containing vertices of degrees 3, 5, 7, and 12, respectively.
Moreover, |V1(HDN1[n])| = 6n2–12n+6, |V2(HDN1[n])| = 6, |V3(HDN1[n])| = 6n – 12, and |V4(HDN1[n])| = 3n2 – 9n + 7.
Similarly, the edge set E of HDN1[n] can be divided into eight disjoint subsets:
Also,
Now, we have
Some degree-based topologcal indices of HDN1[n] are given in the following proposition.
Proposition 1
For the hex-derived network HDN1[n], we have
M1(HDN1[n]) = 486n2 – 966n + 480.
M2(HDN1[n]) = 1944n2 – 4596n + 2718.
Proof
Consider the M-polynomial of HDN1[1], which is
Then
Now, we go for indices.
M1(HDN1[n]) = Dx + Dy(f(x, y))|x=y=1 = 486n2 − 966n + 480
M2(HDN1[n]) = DxDy(f(x, y))|x=y=1 = 1944n2 − 4596n + 2718.
Theorem 2
The M-polynomial of HDN2[n] is
Proof
Consider the hexagonal mesh HDN2[n], where n > 3. The vertex set of HDN2[n] has the following four partitions.
and
Also
Moreover we divide the edge set of HDN2[n] into the following ten partitions:
and
Also, we have
and
Now, by the definition of the M-polynomial, we have
Now we compute some degree-based topological indices of the hexagonal mesh.
Proposition 2
Forwe have
M1(HDN2[n]) = 648n2 − 1500n + 852.
M2(HDN2[n]) = 2916n2 − 7566n + 4764.
Proof
Let
Then
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].
![Figure 6 The plot for the M-polynomial of HDN1[1]](/document/doi/10.1515/phys-2018-0054/asset/graphic/j_phys-2018-0054_fig_006.jpg)
The plot for the M-polynomial of HDN1[1]
![Figure 7 The plot for the M-polynomial of HDN2[1]](/document/doi/10.1515/phys-2018-0054/asset/graphic/j_phys-2018-0054_fig_007.jpg)
The plot for the M-polynomial of HDN2[1]
Acknowledgement
This research is supported by Higher Education Commission of Pakistan.
Autor Contributions: All authors contributed equally in writing this paper.
Conflict of Interest
Conflict of Interests: The authors declare no conflict of interest.
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- Characteristics of microscopic pore-throat structure of tight oil reservoirs in Sichuan Basin measured by rate-controlled mercury injection
- An activity window model for social interaction structure on Twitter
- Transient thermal regime trough the constitutive matrix applied to asynchronous electrical machine using the cell method
- On the zagreb polynomials of benzenoid systems
- Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance
- The Greek parameters of a continuous arithmetic Asian option pricing model via Laplace Adomian decomposition method
- Quantifying the global solar radiation received in Pietermaritzburg, KwaZulu-Natal to motivate the consumption of solar technologies
- Sturm-Liouville difference equations having Bessel and hydrogen atom potential type
- Study on the response characteristics of oil wells after deep profile control in low permeability fractured reservoirs
- Depiction and analysis of a modified theta shaped double negative metamaterial for satellite application
- An attempt to geometrize electromagnetism
- Structure of traveling wave solutions for some nonlinear models via modified mathematical method
- Thermo-convective instability in a rotating ferromagnetic fluid layer with temperature modulation
- Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations
- Effect of magnetic field and heat source on Upper-convected-maxwell fluid in a porous channel
- Physical cues of biomaterials guide stem cell fate of differentiation: The effect of elasticity of cell culture biomaterials
- Shooting method analysis in wire coating withdrawing from a bath of Oldroyd 8-constant fluid with temperature dependent viscosity
- Rank correlation between centrality metrics in complex networks: an empirical study
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering
- Modeling of electric and heat processes in spot resistance welding of cross-wire steel bars
- Dynamic characteristics of triaxial active control magnetic bearing with asymmetric structure
- Design optimization of an axial-field eddy-current magnetic coupling based on magneto-thermal analytical model
- Thermal constitutive matrix applied to asynchronous electrical machine using the cell method
- Temperature distribution around thin electroconductive layers created on composite textile substrates
- Model of the multipolar engine with decreased cogging torque by asymmetrical distribution of the magnets
- Analysis of spatial thermal field in a magnetic bearing
- Use of the mathematical model of the ignition system to analyze the spark discharge, including the destruction of spark plug electrodes
- Assessment of short/long term electric field strength measurements for a pilot district
- Simulation study and experimental results for detection and classification of the transient capacitor inrush current using discrete wavelet transform and artificial intelligence
- Magnetic transmission gear finite element simulation with iron pole hysteresis
- Pulsed excitation terahertz tomography – multiparametric approach
- Low and high frequency model of three phase transformer by frequency response analysis measurement
- Multivariable polynomial fitting of controlled single-phase nonlinear load of input current total harmonic distortion
- Optimal design of a for middle-low-speed maglev trains
- Eddy current modeling in linear and nonlinear multifilamentary composite materials
- The visual attention saliency map for movie retrospection
- AC/DC current ratio in a current superimposition variable flux reluctance machine
- Influence of material uncertainties on the RLC parameters of wound inductors modeled using the finite element method
- Cogging force reduction in linear tubular flux switching permanent-magnet machines
- Modeling hysteresis curves of La(FeCoSi)13 compound near the transition point with the GRUCAD model
- Electro-magneto-hydrodynamic lubrication
- 3-D Electromagnetic field analysis of wireless power transfer system using K computer
- Simplified simulation technique of rotating, induction heated, calender rolls for study of temperature field control
- Design, fabrication and testing of electroadhesive interdigital electrodes
- A method to reduce partial discharges in motor windings fed by PWM inverter
- Reluctance network lumped mechanical & thermal models for the modeling and predesign of concentrated flux synchronous machine
- Special Issue Applications of Nonlinear Dynamics
- Study on dynamic characteristics of silo-stock-foundation interaction system under seismic load
- Microblog topic evolution computing based on LDA algorithm
- Modeling the creep damage effect on the creep crack growth behavior of rotor steel
- Neighborhood condition for all fractional (g, f, n′, m)-critical deleted graphs
- Chinese open information extraction based on DBMCSS in the field of national information resources
- 10.1515/phys-2018-0079
- CPW-fed circularly-polarized antenna array with high front-to-back ratio and low-profile
- Intelligent Monitoring Network Construction based on the utilization of the Internet of things (IoT) in the Metallurgical Coking Process
- Temperature detection technology of power equipment based on Fiber Bragg Grating
- Research on a rotational speed control strategy of the mandrel in a rotary steering system
- Dynamic load balancing algorithm for large data flow in distributed complex networks
- Super-structured photonic crystal fiber Bragg grating biosensor image model based on sparse matrix
- Fractal-based techniques for physiological time series: An updated approach
- Analysis of the Imaging Characteristics of the KB and KBA X-ray Microscopes at Non-coaxial Grazing Incidence
- Application of modified culture Kalman filter in bearing fault diagnosis
- Exact solutions and conservation laws for the modified equal width-Burgers equation
- On topological properties of block shift and hierarchical hypercube networks
- Elastic properties and plane acoustic velocity of cubic Sr2CaMoO6 and Sr2CaWO6 from first-principles calculations
- A note on the transmission feasibility problem in networks
- Ontology learning algorithm using weak functions
- Diagnosis of the power frequency vacuum arc shape based on 2D-PIV
- Parametric simulation analysis and reliability of escalator truss
- A new algorithm for real economy benefit evaluation based on big data analysis
- Synergy analysis of agricultural economic cycle fluctuation based on ant colony algorithm
- Multi-level encryption algorithm for user-related information across social networks
- Multi-target tracking algorithm in intelligent transportation based on wireless sensor network
- Fast recognition method of moving video images based on BP neural networks
- Compressed sensing image restoration algorithm based on improved SURF operator
- Design of load optimal control algorithm for smart grid based on demand response in different scenarios
- Face recognition method based on GA-BP neural network algorithm
- Optimal path selection algorithm for mobile beacons in sensor network under non-dense distribution
- Localization and recognition algorithm for fuzzy anomaly data in big data networks
- Urban road traffic flow control under incidental congestion as a function of accident duration
- Optimization design of reconfiguration algorithm for high voltage power distribution network based on ant colony algorithm
- Feasibility simulation of aseismic structure design for long-span bridges
- Construction of renewable energy supply chain model based on LCA
- The tribological properties study of carbon fabric/ epoxy composites reinforced by nano-TiO2 and MWNTs
- A text-Image feature mapping algorithm based on transfer learning
- Fast recognition algorithm for static traffic sign information
- Topical Issue: Clean Energy: Materials, Processes and Energy Generation
- An investigation of the melting process of RT-35 filled circular thermal energy storage system
- Numerical analysis on the dynamic response of a plate-and-frame membrane humidifier for PEMFC vehicles under various operating conditions
- Energy converting layers for thin-film flexible photovoltaic structures
- Effect of convection heat transfer on thermal energy storage unit
