Numerous graph energies of regular subdivision graph and complete graph

Imrana Kousar; Saima Nazeer; Abid Mahboob; Sana Shahid; Yu-Pei Lv; Imrana Kousar; Saima Nazeer; Abid Mahboob; Sana Shahid; Yu-Pei Lv

doi:10.3934/math.2021491

AIMS Mathematics

2021, Volume 6, Issue 8: 8466-8476. doi: 10.3934/math.2021491

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Research article

Numerous graph energies of regular subdivision graph and complete graph

1.
Department of Mathematics, Lahore College for Women University, Lahore-Pakistan
2.
Department of Mathematics, Division of Science and Technology, University of Education, Lahore-Pakistan
3.
Department of Mathematics, Huzhou University, Huzhou 313000, China

Received: 27 April 2021 Accepted: 19 May 2021 Published: 02 June 2021
MSC : 14H50, 14H20, 32S15

The graph energy $E(G)$ of a simple graph $G$ is sum of its absolute eigenvalues where eigenvalues of adjacency matrix $A(G)$ are referred as eigenvalues of graph $G$ . Depends upon eigenvalues of different graph matrices, several graph energies has been observed recently such as maximum degree energy, Randi $\acute{c}$ energy, sum-connectivity energy etc. Depending on the definition of a graph matrix, the graph energy can be easily determined. This article contains upper bounds of several graph energies of $s$ -regular subdivision graph $S(G)$ . Also various graph energies of complete graph are mentioned in this article.

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Citation: Imrana Kousar, Saima Nazeer, Abid Mahboob, Sana Shahid, Yu-Pei Lv. Numerous graph energies of regular subdivision graph and complete graph[J]. AIMS Mathematics, 2021, 6(8): 8466-8476. doi: 10.3934/math.2021491

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Abstract

1. Introduction

The study on flow of fluids which are electrically conducting is known as magnetohydrodynamics (MHD). The magnetohydrodynamics have important applications in the polymer industry and engineering fields (Garnier ^[1]). Heat transfer caused by hydromagnetism was discussed by Chakrabarthi and Gupta ^[2]. Using an exponentially shrinking sheet, Nadeem et al. ^[3] investigated the MHD flow of a Casson fluid. Krishnendu Bhattacharyya ^[4] examined the effect of thermal radiation on MHD stagnation-point Flow over a Stretching Sheet.

Mixed convection magnetohydrodynamics flow is described by Ishak on a vertical and on a linearly stretching sheet ^[5,6,7]. Hayat et al. ^[10] examine a mixed convection flow within a stretched sheet of Casson nanofluid. Subhas Abel and Monayya Mareppa, examine magnetohydrodynamics flow on a vertical plate ^[11]. Shen et al. ^[12], examined a vertical stretching sheet which was non-linear. Ishikin Abu Bakar ^[13] investigates how boundary layer flow is affected by slip and convective boundary conditions over a stretching sheet. A vertical plate oscillates with the influence of slip on a free convection flow of a Casson fluid ^[14].

Nasir Uddin et al. ^[15] used a Runge-Kutta sixth-order integration method. Barik et al. ^[16] implicit finite distinction methodology of Crank Sir Harold George Nicolson sort Raman and Kumar ^[17] utilized an exact finite distinction theme of DuFort–Frankel. Mondal et al. ^[18] used a numerical theme over the whole vary of physical parameters. With the laplace transform method, we can determine the magnetohydrodynamic flow of a viscous fluid ^[19]. Thamizh Suganya et al. ^[20] obtained that the MHD for the free convective flow of fluid is based on coupled non-linear differential equations. In this study, the analytical approximation of concentration profiles in velocity, temperature and concentration using homotopy perturbation method (HPM).

2. Mathematical construction of the problem

The governing differential equations in dimensionless form ^[19] as follows:

$\begin{align} \frac{d^2 u}{dy^2} - Hu + Gr \theta + Gm \phi = 0, \end{align}$

(2.1)

$\begin{align} \frac{1}{F} \frac{d^2 \theta}{dy^2} = 0, \end{align}$

(2.2)

$\begin{align} \frac{1}{Sc} \frac{d^2 \phi}{dy^2} - Sr \frac{\partial^2 \phi}{\partial y^2} - \gamma \phi = 0. \end{align}$

(2.3)

The dimensionless boundary conditions given by:

$\begin{equation} u = 0, \ \theta = 1, \phi = 0 \ \ at \ \ y = 0 \end{equation}$

(2.4)

and

$\begin{equation} u = 0, \ \theta = 0, \phi = 0 \ \ at \ \ y \to \infty. \end{equation}$

(2.5)

3. Solutions of steady-state concentration profile using the Homotopy Perturbation Method

He ^[21,22] established the homotopy perturbation method, which waives the requirement of small parameters. Many researchers have used HPM to obtain approximate analytical solutions for many non-linear engineering dynamical systems ^[23,24]. The basic concept of the HPM as follows:

$\begin{align} \frac{d^2 u}{dy^2} - Hu + Gr \theta + Gm \phi = 0 \end{align}$

(3.1)

$\begin{align} \frac{1}{F} \frac{d^2 \theta}{dy^2} = 0 \end{align}$

(3.2)

$\begin{align} \frac{1}{Sc} \frac{d^2 \phi}{dy^2} - Sr \frac{\partial^2 \phi}{\partial y^2} - \gamma \phi = 0 \end{align}$

(3.3)

with initial and boundary conditions given by:

$\begin{align} y = 0 \ \ &at \ u = 0, \ \theta = 1, C = 1 \\ y \to \infty \ \ &as \ u = 0, \ \theta = 1, \phi = 0. \end{align}$

(3.4)

Homotopy for the above Eqs (3.1) to (3.4) can be constructed as follows:

$\begin{align} (1 - p) \left[\frac{d^2 u}{dy^2} - Hu + Gr \theta + Gm \phi\right] + p \left[\frac{d^2 u}{dy^2} - Hu + Gr \theta + Gm \phi\right] = 0 \end{align}$

(3.5)

$\begin{align} (1 - p) \left[\frac{1}{F} \frac{d^2 \theta}{dy^2} - \theta\right] + p \left[\frac{1}{F} \frac{d^2 \theta}{dy^2} - \theta + \theta\right] = 0 \end{align}$

(3.6)

$\begin{align} (1 - p) \left[\frac{1}{Sc} \frac{d^2 \phi}{dy^2} - \gamma \phi \right] + p \left[\frac{1}{Sc} \frac{d^2 \phi}{dy^2} - Sr \frac{\partial^2 \theta}{\partial y^2} - \gamma \phi \right] = 0 \end{align}$

(3.7)

The approximate solution of the Eqs (3.5) to (3.7) are

$\begin{align} u & = u_0 + pu_1 + p^2 u_2 + p^3 u_3 + ... \end{align}$

(3.8)

$\begin{align} \theta & = \theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ... \end{align}$

(3.9)

$\begin{align} \phi & = \phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ... \end{align}$

(3.10)

Substitution Eqs (3.5) to (3.7) in Eqs (3.8) to (3.10) respectively. We obtain the following equations

$\begin{align} (1 - p) &\left[\frac{d^2 (u_0 + pu_1 + p^2 u_2 + p^3 u_3 + ...)}{dy^2} - H(u_0 + pu_1 + p^2 u_2 + p^3 u_3 + ...) \right. \\ &\left.+ Gr (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...) + Gm (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...)\right] \\ &+ p\left[\frac{d^2 (u_0 + pu_1 + p^2 u_2 + p^3 u_3 + ...)}{dy^2} - H(u_0 + pu_1 + p^2 u_2 + p^3 u_3 + ...)\right. \\ &\left.+ Gr (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...) + Gm (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...)\right] = 0 \end{align}$

(3.11)

and

$\begin{align} (1 - p) &\left[\frac{1}{F} \frac{d^2 (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...)}{dy^2} - (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...)\right] \\ &+ p \left[\frac{1}{F} \frac{d^2 (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...)}{dy^2} - (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...) \right. \\ &\left.+ (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...)\right] = 0, \end{align}$

(3.12)

and

$\begin{align} (1 - p) &\left[\frac{1}{Sc} \frac{d^2 (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...)}{dy^2} - \gamma (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...)\right] \\ &+ p \left[\frac{1}{Sc} \frac{d^2 (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...)}{dy^2} - Sr \frac{\partial^2 (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...)}{\partial y^2} \right. \\ &\left.- \gamma (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...) \right] = 0. \end{align}$

(3.13)

Equating the coefficient of $p$ on both sides, we get the following equations

$\begin{align} P^0 &: \frac{d^2 u_0}{dy^2} - Hu_0 + Gr \theta_0 + Gm \phi_0 = 0; \end{align}$

(3.14)

$\begin{align} p^0 &: \frac{1}{F} \frac{d^2 \theta_0}{dy^2} - \theta_0 = 0; \end{align}$

(3.15)

$\begin{align} P^1 &: \frac{1}{F} \frac{d^2 \theta_0}{dy^2} - \theta_0 + \theta_1 = 0; \end{align}$

(3.16)

$\begin{align} P^1 &: \frac{1}{Sc} \frac{d^2 \phi_0}{dy^2} - \gamma \phi_0 = 0; \end{align}$

(3.17)

$\begin{align} P^1 &: \frac{1}{Sc} \frac{d^2 \phi_0}{dy^2} - Sr \frac{\partial^2 \theta_0}{\partial y^2} - \gamma \phi_0 = 0. \end{align}$

(3.18)

The boundary conditions are

$\begin{align} u_0 = 0, \ \theta_0 = 1, \phi_0 = 1 \ \ at \ \ y = 0 \\ u_0 = 0, \ \theta_0 = 1, \phi_0 = 1 \ \ at \ \ y \to \infty. \end{align}$

(3.19)

and

$\begin{align} u_1 = 0, \ \theta_1 = 0, \phi_1 = 0 \ \ at \ \ y = 0 \\ u_1 = 0, \ \theta_1 = 0, \phi_1 = 0 \ \ at \ \ y \to \infty. \end{align}$

(3.20)

Solving the Eqs (3.9)–(3.14), we obtain

$\begin{align} u_0(t) & = \frac{Gr}{\sqrt{\frac{1}{F}} + H} \left[e^{-y \sqrt{\frac{1}{F}}} - e^{-y \sqrt{H}}\right] + \frac{Gm}{\sqrt{\gamma Sc + H}} \left[e^{-y \sqrt{\gamma Sc}} - e^{-y \sqrt{H}}\right]; \end{align}$

(3.21)

$\begin{align} \theta_0(y) & = e^{-y \sqrt{\frac{1}{F}}}; \end{align}$

(3.22)

$\begin{align} \phi_0(y) & = e^{-y \sqrt{\gamma Sc}}; \end{align}$

(3.23)

$\begin{align} \phi_1(y) & = \frac{Sc Sr}{\sqrt{F} + F \gamma Sc} \left[e^{-y \sqrt{\frac{1}{F}}} - e^{-y \sqrt{\gamma Sc}}\right]. \end{align}$

(3.24)

Considering the iteration, we get,

$\begin{align} u(t) & = \frac{Gr}{\sqrt{\frac{1}{F}} + H} \left[e^{-y \sqrt{\frac{1}{F}}} - e^{-y \sqrt{H}}\right] + \frac{Gm}{\sqrt{\gamma Sc + H}} \left[e^{-y \sqrt{\gamma Sc}} - e^{-y \sqrt{H}}\right]; \end{align}$

(3.25)

$\begin{align} \theta(y) & = e^{-y \sqrt{F}}; \end{align}$

(3.26)

$\begin{align} \phi(y) & = e^{-y \sqrt{\gamma Sc}} + \frac{Sc Sr}{\sqrt{F} + F \gamma Sc} \left[e^{-y \sqrt{\frac{1}{F}}} - e^{-y \sqrt{\gamma Sc}}\right]. \end{align}$

(3.27)

4. Approximate analytical solutions for the skiin friction, Nusselt and Sherwood numbers

From the Eqs (3.25)–(3.27), we obtain

$\begin{array}{l} C_f = - \left(\frac{\partial u}{\partial y}\right)_{y = 0} \\ = \frac{\left(Gm \sqrt{\gamma Sc} - Gm \sqrt{H} + Gr(\gamma Sc + H) \sqrt{1 + \frac{R}{Pr}} + Gm H \sqrt{\gamma Sc} + (-Gm - Gr) H^{\frac{3}{2}} - \gamma Gr \sqrt{H} Sc\right)}{\left(\sqrt{1 + \frac{R}{Pr} + H} (\gamma Sc + H)\right)}; \end{array}$

(4.1)

$\begin{align} N_u & = - \left(\frac{\partial \theta}{\partial y}\right)_{y = 0} = \sqrt{\frac{1 + R}{Pr}}; \end{align}$

(4.2)

$\begin{array}{l} S_h = - \left(\frac{\partial \phi}{\partial y}\right)_{y = 0} \\= \frac{Sc Sr (1 + R) \sqrt{1 + \frac{R}{Pr}} - \sqrt{\gamma Sc} \left((-R - 1) \sqrt{1 + \frac{R}{Pr}} + ((1 + R)Sc - Pr \gamma) Sc \right)}{(1 + R) \sqrt{1 + \frac{R}{Pr}} Pr \gamma Sc} \end{array}$

(4.3)

5. Results and discussion

The combined impacts of transient MHD free convective flows of an incompressible viscous fluid through a vertical plate moving with uniform motion and immersed in a porous media are examined using an exact approach. The approximate analytical expressions for the velocity $u$ , temperature $\theta$ , and concentration profile $\phi$ are solved by using the homotopy perturbation method for fixed values of parameters is graphically presented.

Velocity takes time at first, and for high values of y, it takes longer, and the velocity approaches zero as time increases. The velocity of fluid rises with Gr increasing, as exposed in Figure 1.

Figure 1. An illustration of velocity profiles for different values of Gr.

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AIMS Mathematics

Numerous graph energies of regular subdivision graph and complete graph

Related Papers:

Abstract

1. Introduction

2. Mathematical construction of the problem

3. Solutions of steady-state concentration profile using the Homotopy Perturbation Method

4. Approximate analytical solutions for the skiin friction, Nusselt and Sherwood numbers

5. Results and discussion

6. Conclusions

Conflict of interest

Acknowledgement

References

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Abstract

1. Introduction

2. Mathematical construction of the problem

3. Solutions of steady-state concentration profile using the Homotopy Perturbation Method

4. Approximate analytical solutions for the skiin friction, Nusselt and Sherwood numbers

5. Results and discussion

6. Conclusions

Conflict of interest

Acknowledgement

References