The extremal unicyclic graphs with given diameter and minimum edge revised Szeged index

Shengjie He; Qiaozhi Geng; Rong-Xia Hao; Shengjie He; Qiaozhi Geng; Rong-Xia Hao

doi:10.3934/math.20231342

AIMS Mathematics

2023, Volume 8, Issue 11: 26301-26327. doi: 10.3934/math.20231342

Previous Article Next Article

Research article

The extremal unicyclic graphs with given diameter and minimum edge revised Szeged index

1.
School of Science, Tianjin University of Commerce, Tianjin 300134, China
2.
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

Received: 22 July 2023 Revised: 21 August 2023 Accepted: 28 August 2023 Published: 14 September 2023
MSC : 05C12, 05C90

Let $H$ be a connected graph. The edge revised Szeged index of $H$ is defined as $Sz^{\ast}_{e}(H) = \sum\limits_{e = uv\in E_H}(m_{u}(e|H)+\frac{m_{0}(e|H)}{2})(m_{v}(e|H)+\frac{m_{0}(e|H)}{2})$ , where $m_{u}(e|H)$ (resp., $m_{v}(e|H)$ ) is the number of edges whose distance to vertex $u$ (resp., $v$ ) is smaller than to vertex $v$ (resp., $u$ ), and $m_{0}(e|H)$ is the number of edges equidistant from $u$ and $v$ . In this paper, the extremal unicyclic graphs with given diameter and minimum edge revised Szeged index are characterized.

Keywords:

Citation: Shengjie He, Qiaozhi Geng, Rong-Xia Hao. The extremal unicyclic graphs with given diameter and minimum edge revised Szeged index[J]. AIMS Mathematics, 2023, 8(11): 26301-26327. doi: 10.3934/math.20231342

Related Papers:

[1]	Min Li, Ke Chen, Yunqing Bai, Jihong Pei . Skeleton action recognition via graph convolutional network with self-attention module. Electronic Research Archive, 2024, 32(4): 2848-2864. doi: 10.3934/era.2024129
[2]	Wangwei Zhang, Menghao Dai, Bin Zhou, Changhai Wang . MCADFusion: a novel multi-scale convolutional attention decomposition method for enhanced infrared and visible light image fusion. Electronic Research Archive, 2024, 32(8): 5067-5089. doi: 10.3934/era.2024233
[3]	Bingsheng Li, Na Li, Jianmin Ren, Xupeng Guo, Chao Liu, Hao Wang, Qingwu Li . Enhanced spectral attention and adaptive spatial learning guided network for hyperspectral and LiDAR classification. Electronic Research Archive, 2024, 32(7): 4218-4236. doi: 10.3934/era.2024190
[4]	Hui Xu, Jun Kong, Mengyao Liang, Hui Sun, Miao Qi . Video behavior recognition based on actional-structural graph convolution and temporal extension module. Electronic Research Archive, 2022, 30(11): 4157-4177. doi: 10.3934/era.2022210
[5]	Huimin Qu, Haiyan Xie, Qianying Wang . Multi-convolutional neural network brain image denoising study based on feature distillation learning and dense residual attention. Electronic Research Archive, 2025, 33(3): 1231-1266. doi: 10.3934/era.2025055
[6]	Kun Zheng, Li Tian, Zichong Li, Hui Li, Junjie Zhang . Incorporating eyebrow and eye state information for facial expression recognition in mask-obscured scenes. Electronic Research Archive, 2024, 32(4): 2745-2771. doi: 10.3934/era.2024124
[7]	Bojian Chen, Wenbin Wu, Zhezhou Li, Tengfei Han, Zhuolei Chen, Weihao Zhang . Attention-guided cross-modal multiple feature aggregation network for RGB-D salient object detection. Electronic Research Archive, 2024, 32(1): 643-669. doi: 10.3934/era.2024031
[8]	Jiange Liu, Yu Chen, Xin Dai, Li Cao, Qingwu Li . MFCEN: A lightweight multi-scale feature cooperative enhancement network for single-image super-resolution. Electronic Research Archive, 2024, 32(10): 5783-5803. doi: 10.3934/era.2024267
[9]	Wangwei Zhang, Hao Sun, Bin Zhou . TBRAFusion: Infrared and visible image fusion based on two-branch residual attention Transformer. Electronic Research Archive, 2025, 33(1): 158-180. doi: 10.3934/era.2025009
[10]	Jingqian Xu, Ma Zhu, Baojun Qi, Jiangshan Li, Chunfang Yang . AENet: attention efficient network for cross-view image geo-localization. Electronic Research Archive, 2023, 31(7): 4119-4138. doi: 10.3934/era.2023210

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.

[1]	A. Bondy, U. S. R. Murty, Graph theory, New York: Springer, 2008. http://dx.doi.org/10.1007/978-1-84628-970-5
[2]	X. Cai, B. Zhou, Edge Szeged index of unicyclic graphs, MATCH Commun. Math. Comput. Chem., 63 (2010), 133–144.
[3]	P. Dankelmanna, I. Gutman, S. Mukwembi, H. C. Swart, The edge-Wiener index of a graph, Discrete Math., 309 (2009), 3452–3457. http://dx.doi.org/10.1016/j.disc.2008.09.040 doi: 10.1016/j.disc.2008.09.040
[4]	H. Dong, B. Zhou, C. Trinajstić, A novel version of the edge-Szeged index, Croat. Chem. Acta., 84 (2011), 543–545. http://dx.doi.org/10.5562/cca1889 doi: 10.5562/cca1889
[5]	I. Gutman, A. R. Ashrafi, The edge version of the Szeged index, Croat. Chem. Acta., 81 (2008), 263–266.
[6]	S. He, R. X. Hao, Y. Q. Feng, On the edge-Szeged index of unicyclic graphs with perfect matchings, Discrete Appl. Math., 284 (2020), 207–223. http://dx.doi.org/10.1016/j.dam.2020.03.033 doi: 10.1016/j.dam.2020.03.033
[7]	S. He, R. X. Hao, A. Yu, On extremal cacti with respect to the edge Szeged index and edge-vertex Szeged index, Filomat, 32 (2018), 4069–4078. http://dx.doi.org/10.2298/FIL1811069H doi: 10.2298/FIL1811069H
[8]	P. Khadikar, P. Kale, N. Deshpande, S. Karmarkar, V. Agrawal, Szeged indices of hexagonal chains, MATCH Commun. Math. Comput. Chem., 43 (2001), 7–15.
[9]	J. Li, A relation between the edge Szeged index and the ordinary Szeged index, MATCH Commun. Math. Comput. Chem., 70 (2013), 621–625.
[10]	X. Li, Y. Shi, I. Gutman, Graph energy, New York: Springer, 2012. http://dx.doi.org/10.1007/978-1-4614-4220-2
[11]	X. Li, Y. Shi, L. Wang, On a relation between Randić index and algebraic connectivity, MATCH Commun. Math. Comput. Chem., 68 (2011), 843–849.
[12]	H. Liu, X. Pan, On the Wiener index of trees with fixed diameter, MATCH Commun. Math. Comput. Chem., 60 (2008), 85–94.
[13]	M. Liu, S. Wang, Cactus graphs with minimum edge revised Szeged index, Discrete Appl. Math., 247 (2018), 90–96. http://dx.doi.org/10.1016/j.dam.2018.03.037 doi: 10.1016/j.dam.2018.03.037
[14]	Y. Liu, A. Yu, M. Lu, R. X. Hao, On the Szeged index of unicyclic graphs with given diameter, Discrete Appl. Math., 233 (2017), 118–130. http://dx.doi.org/10.1016/j.dam.2017.08.009 doi: 10.1016/j.dam.2017.08.009
[15]	M. J. Nadjafi-Arani, H. Khodashenas, A. R. Ashrafi, Relationship between edge Szeged and edge Wiener indices of graphs, Glas. Mat., 47 (2012), 21–29. http://dx.doi.org/10.3336/gm.47.1.02 doi: 10.3336/gm.47.1.02
[16]	M. Randić, On generalization of Wiener index for cyclic structures, Acta Chim. Slov., 49 (2002), 483–496.
[17]	C. Ren, J. Shi, On the Wiener index of unicyclic graphs with fixed diameter, Journal of East China University of Science and Technology, 39 (2013), 768–772.
[18]	S. Tan, The minimum Wiener index of unicyclic graphs with a fixed diameter, J. Appl. Math. Comput., 56 (2018), 93–114. http://dx.doi.org/10.1007/s12190-016-1063-2 doi: 10.1007/s12190-016-1063-2
[19]	G. Wang, S. Li, D. Qi, H. Zhang, On the edge-Szeged index of unicyclic graphs with given diameter, Appl. Math. Comput., 336 (2018), 94–106. http://dx.doi.org/10.1016/j.amc.2018.04.077 doi: 10.1016/j.amc.2018.04.077
[20]	M. Wang, M. Liu, The lower bound of edge revised Szeged index of unicyclic graphs with given diameter, Advances in Mathematics (China), 52 (2023), 25–45. http://dx.doi.org/10.11845/sxjz.2021043b doi: 10.11845/sxjz.2021043b
[21]	H. Wiener, Structral determination of paraffin boiling points, J. Am. Chem. Soc., 69 (1947), 17–20. http://dx.doi.org/10.1021/ja01193a005 doi: 10.1021/ja01193a005
[22]	A. Yu, K. Peng, R. X. Hao, J. Fu, Y. Wang, On the revised Szeged index of unicyclic graphs with given diameter, Bull. Malays. Math. Sci. Soc., 43 (2020), 651–672. http://dx.doi.org/10.1007/s40840-018-00706-4 doi: 10.1007/s40840-018-00706-4

AIMS Mathematics

The extremal unicyclic graphs with given diameter and minimum edge revised Szeged index

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

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog