######
(*β ,α*)*−*Connectivity Index of Graphs

1 Introduction Let G = ( V,E ) be a simple graph with n = | V | vertices and m = | E |edges. As usual, n is said to be an order and m the size of G . The subdivision graph S ( G ) is the graph obtained from G by replacing each edge by a path of length 2. The line graph L ( G ) of G is the graph whose vertex set is E ( G ) in which two vertices are adjacent if and only if they are adjacent in G . The tadpole graph T n,k is the graph obtained by joining a cycle of n vertices with a path of length k . The cartesian product G × H of graphs

###### Designing optimal trajectories for a skimmer ship to clean, recover and prevent the oil spilled on the sea from reaching the coast

and analyse the behaviour of the solution regarding some key parameters (as the pumping power of the skimmer). Finally, some conclusions are made in Section 4 2 Materials and Methods 2.1 Eulerian Mathematical Model Mathematical modelling of the transport and diffusion of an oil spill in the sea is of high interest to remediate the environmental impact (e.g., [ 24 , 25 , 26 ]). We have developed an Eulerian model for the case where the density of the pollutant is smaller than that of one of the sea water (so that it remains at the surface) and

###### Poisson and symplectic reductions of 4–DOF isotropic oscillators. The van der Waals system as benchmark

{\kern 1pt} {\rm{ }}\cos E){\kern 1pt} dE. \end{array}$$ (25) Completing the functions of the momenta given above (23) , it is convenient to introduce two more state functions w = U 1 / L , z = U 3 / L . $$\begin{array}{} \displaystyle w=U_1/L,\qquad z=U_3/L. \end{array}$$ (26) The previous results may be summarized in the following theorem: Theorem 3. By composing the three symplectic transformations ( q 1 , q 2 , q 3 , q 4 Q 1 , Q 2 , Q 3 , Q 4 ) Projective Euler → ( ρ , ϕ , θ , ψ P , Φ , Θ , Ψ ) ↓ ↓ P A ( ℓ , u 1 , g , u 3 L , U 1 , G , U 3

###### Computing the two first probability density functions of the random Cauchy-Euler differential equation: Study about regular-singular points

Advanced Engineering Mathematics Wiley & Sons Boston [9] Kwok Sau Fa, Wang K.G. (2012). Integro-differential equation for joint probability density in phase space associated with continuous-time random walk. Physica A. 391, no. 15, 3858–3864. 10.1016/j.physa.2012.03.013 Kwok Sau Fa. Wang K.G. 2012 Integro-differential equation for joint probability density in phase space associated with continuous-time random walk Physica A. 391 15 3858 3864 10.1016/j.physa.2012.03.013 [10] Santos, L.T., Dorini, F.A., Cunha, M.C.C. (2010). The probability density function to the

###### Homoclinic and Heteroclinic Motions in Economic Models with Exogenous Shocks

Decoster G. P. Labys W. C. Mitchell D. W. 1992 Evidence of chaos in commodity futures prices Journal of Futures Markets 12 291 305 10.1002/fut.3990120305 [28] A. Wei and R. M. Leuthold. (1998), Long Agricultural Futures Prices: ARCH, Long Memory or Chaos Processes? OFOR Paper 98-03. University of Illinois at Urbana-Champaign, Urbana. Wei A. Leuthold R. M. 1998 Long Agricultural Futures Prices: ARCH, Long Memory or Chaos Processes? OFOR Paper 98-03 University of Illinois at Urbana-Champaign Urbana [29] E. Panas and V. Ninni. (2000), Are oil markets chaotic? A non

###### Symmetry Reductions for a Generalized Fifth Order KdV Equation

-Stokes equations of fluid dynamics [ 8 , 24 ] or the Koterweg de Vries equations (KdV) which model waves on shallow water surfaces [ 17 , 25 ]. In the present paper we are going to focus our attention in Koterweg de Vries equations. The KdV equation u t + λ u u x + μ u x x x = 0 $$ \begin{equation} u_t+\lambda u u_x + \mu u_{xxx}=0 \end{equation} $$ (1) was arised to model shallow water waves with weak nonlinearities and it is probably the most studied nonlinear evolution equation due to its wide applicability. The KdV

###### Wall Properties and Slip Consequences on Peristaltic Transport of a Casson Liquid in a Flexible Channel with Heat Transfer

transport with heat transfer ComptesRendusM茅canique 335 7 369 373 [24] T. Hayat, M. Javed, S. Asghar and A. A. Hendi, (2013), Wall properties and heat transfer analysis of the peristaltic motion in a power-law fluid, International journal for Numerical methods in fluids, 71, No. 1, 65-79. 10.1002/fld.3647 Hayat T. Javed M. Asghar S. Hendi A.A. 2013 Wall properties and heat transfer analysis of the peristaltic motion in a power-law fluid International journal for Numerical methods in fluids 71 1 65 79 [25] P. Lakshminarayana, S. Sreenadh and G. Sucharitha, (2015

###### Research on relationship between tourism income and economic growth based on meta-analysis

economic growth, Economic and Management Review, Vol. 33, pp. 108-112. 10.13962/j.cnki.37-1486/f.2017.03.013 Ai Min D. Hui L. 2017 An empirical study on the contribution of inbound tourism to regional economic growth Economic and Management Review 33 108 112 10.13962/j.cnki.37-1486/f.2017.03.013 [12] X. Wen Qing and M. Ya Ni. (2011), The empirical study on the impact of tourism revenue on regional economy, Business Era, pp. 120-121. 10.3969/j.issn.1002-5863.2011.25.052 Wen Qing X. Ya Ni M. 2011 The empirical study on the impact of tourism revenue on regional economy

###### Contact Impact Forces at Discontinuous 2-DOF Vibroimpact

problems. International Journal of bifurcation and chaos, 16(03), 601-629. 10.1142/S0218127406015015 Kowalczyk P. di Bernardo M. Champneys A. R. Hogan S. J. Homer M. Piiroinen P. T. Kuznetsov Yu A. Nordmark A. 2006 Two-parameter discontinuity-induced bifurcations of limit cycles: classification and open problems International Journal of bifurcation and chaos 16 03 601 629 10.1142/S0218127406015015 [11] di Bernardo, M., Budd, C. J., Champneys, A. R., Kowalczyk, P., Nordmark, A. B., Tost, G. O., & Piiroinen, P. T. (2008), Bifurcations in nonsmooth dynamical systems

###### Dirichlet series and analytical solutions of MHD viscous flow with suction / blowing

F ( 0 ) = 0 , F ′ ( 0 ) = 1 , F ′ ( ∞ ) = 0. $$ \begin{equation} F(0)=0,\quad F'(0)=1,\quad F'(\infty)=0. \end{equation} $$ (16) We introduce two variables ξ and G in the form G ( ξ ) = α F ( η ) a n d ξ = α η , $$ \begin{equation} G(\xi)=\alpha F(\eta) \quad and \quad \xi=\alpha\eta, \end{equation} $$ (17) where α > 0 is an amplification factor. In view of Eq. (17) , the system in Eqs. ( 15 - 16 ) is transformed to the form α 2 G ‴ + 1 2 ( f w α