###### New Complex and Hyperbolic Forms for Ablowitz–Kaup–Newell–Segur Wave Equation with Fourth Order

{array}{} \displaystyle 4{u_{{\rm{xt}}}} + {u_{{\rm{xxxt}}}} + 8{u_x}{u_{{\rm{xy}}}} + 4{u_{{\rm{xx}}}}{u_y} - {\rm{\gamma }}{{\rm{u}}_{{\rm{xx}}}} = 0, \end{array}$$ (1) where γ is a real constant with a non-zero value, by using the sine-Gordon expansion method (SGEM). 2 Fundamental Properties of the SGEM Let us consider the following sine-Gordon equation [ 24 , 25 , 26 ]: u x x − u t t = m 2 sin u , $$\begin{array}{} \displaystyle {u_{xx}} - {u_{tt}} = {m^2}\sin \left( u \right), \end{array}$$ (2) where u = u ( x , t ) and m is a real constant. When we

###### High-accuracy approximation of piecewise smooth functions using the Truncation and Encode approach

(n_{eval}^{-1})$ . In the following, we study the approximation of the smooth function f 1 , which is shown in Tables 8 to 10 . Table 8 Parameters as specified in Table 1 for f 1 , K = 15 and the polygonal rule. ε k = ε varies between 10 -1 and 10 -10 . ε 2 -2 ε n eval ‖ v K − v ^ K ‖ ∞ $||v^{k}- \hat v^K||_\infty$ ‖ v K − v ^ K ‖ 1 $||v^{k}- \hat v^K||_{1}$ | E − Ê K | 1e-01 2.5e-02 1.7000e+01 4.7034e-02 1.2506e-02 1.0395e-03 1e-02 2.5e-03 5.7000e+01 2.4734e-03 9.4241e-04 1.3285e-04

###### Reckoning of the Dissimilar Topological indices of Human Liver

first and second Zagreb indices, have been introduced more than thirty years ago by I. Gutman and Trinajstic [ 10 ]. They defined as: M 1 ( G ) = ∑ u ∈ V ( G ) ( d u ) 2 M 2 ( G ) = ∑ u v ∈ E ( G ) ( d u . d v ) . $$\begin{array}{} \displaystyle ~~M_1(G)=\sum_{u\in V(G)} (d_{u})^{2}\\ \displaystyle M_2(G)=\sum_{uv\in E(G)}(d_u .d_v). \end{array}$$ The Zagreb indices found many applications in QSPR and QSAR studies. For more details on this topological indices we refer to [ 11 , 12 , 13 , 21 , 24 , 25 ,]. There are many topological indices defined on

###### Computing First Zagreb index and F-index of New C-products of Graphs

G incident with it. The partial complement of subdivision graph S ¯ ( G ) $\begin{array}{} \displaystyle \bar S(G) \end{array}$ of a graph G whose vertex set is V ( G ) ∪ E ( G ) where two vertices are adjacent if and only if one is a vertex of G and the other is an edge of G non incident with it. Please refer to [ 17 , 25 ] for unexplained graph theoretic terminology and notation. In theoretical chemistry, the physico-chemical properties of chemical compounds are often modeled by means of molecular-graph-based structure-descriptors which are also

###### Hamilton-connectivity of Interconnection Networks Modeled by a Product of Graphs

the longest path or cycle is required the problem is closely related to well-known hamiltonian problems in graph theory. In the rest of this paper, we will use standard terminology in graphs(see ref.[ 2 ]). It is very difficult to determine that a graph is hamiltonian or not. Readers may refer to [ 4 , 5 , 6 ]. 2 Definitions and Notation We follow [ 2 ] for graph-theoretical terminology and notation not defined here. A graph G = ( V,E ) always means a simple graph(without loops and multiple edges), where V = V ( G ) is the vertex set and E = E ( G

###### A Comparative Study on Haar Wavelet and Hosaya Polynomial for the numerical solution of Fredholm integral equations

{equation}$$ It is worth noting that all the above structure–descriptors are either special cases of, or are simply related to the graph invariant W l , defined as [ 25 , 26 ] (4) W λ = W λ ( G ) = ∑ k ≥ 1 d ( G , k ) k λ $$\begin{equation}\label{w} W_\lambda = W_\lambda(G) = \sum_{k \geq 1} d(G,k)\,k^\lambda \end{equation}$$ where d ( G,k ) is the number of pairs of vertices of the graph G whose distance is k , and where λ is some real (or complex) number. Evidently, W

###### Degree-based indices computation for special chemical molecular structures using edge dividing method

Wang [ 7 ], [ 8 ] and [ 9 ], Xi and Gao [ 10 ], Gao et al. [ 11 ], Gao et al., [ 12 ] and [ 13 ], Gao and Farahani [ 14 ], Farahani and Gao [ 15 ], and Farahani [ 16 ], [ 17 ], [ 18 ], [ 19 ], [ 20 ], [ 21 ], [ 22 ], [ 23 ], [ 24 ] and [ 25 ] for more details). The notations and terminologies that were used but were undefined in this paper can be found in [ 26 ]. All the molecular graphs considered in our paper are simple graphs. Let G be a (molecular) graph with vertex and edge sets being denoted by V ( G ) and E ( G ), respectively. Bollobas and Erdos [ 27

######
A sufficient condition for the existence of a *k*-factor excluding a given *r*-factor

integer and let G be a graph of order n > 8 k 2 − 2( α + 12) k + 3 α + 16 , where α = 3 for odd k and α = 4 for even k. Suppose that kn is even and the minimum degree δ ( G ) of G is at least k. If for any nonadjacent vertices x and y of G, d G ( x ) + d G ( y ) ≥ n + α, then G has a k-factor including a given Hamiltonian cycle . Theorem 11 (Gao, Li, and Li [ 15 ]) Let k ≥ 2 be an integer and let G be a graph of order n > 12( k − 2) 2 + 2(5 − α )( k − 2) − α. Suppose that kn is even, δ ( G ) ≥ k and max { d G ( x ) , d G ( y ) } ≥ n + α

###### Computing topological indices of the line graphs of Banana tree graph and Firecracker graph

edges of G . Line graphs are very useful in mathematical chemistry, but in recent years they were considered very little in chemical graph theory. For further facts about the applications of line graphs in chemistry, we mention the articles [ 6 , 25 , 32 ]. Topological indices are the arithmetical numbers which depends upon the construction of any simple graph. Topological indices are generally classified into three kinds: degree-based indices (see [ 2 , 5 , 7 , 8 , 28 , 29 ]), distance-based indices (see [ 3 , 16 ]), and spectrum-based indices (see [ 11

###### Modified Wavelet Full-Approximation Scheme for the Numerical Solution of Nonlinear Volterra integral and integro-differential Equations

refinement relation of scaling function ϕ ( t ) is given by, ϕ ( t ) = 2 ∑ k = 0 L − 1 h k ϕ ( 2 t − k ) $$\begin{array}{} \displaystyle \phi (t) = \sqrt 2 \mathop \Sigma \limits_{k = 0}^{L - 1} {h_k}\phi (2t - k) \end{array}$$ (1) where ϕ ( t ) is normalized: ∫ − ∞ ∞ ϕ ( t ) d t = 1 $\begin{array}{} \displaystyle \int_{-\infty}^{\infty} \phi(t) dt =1 \end{array}$ . Based on the scaling function ϕ ( t ) , the wavelet function can be written as, The refinement relation of wavelet function ψ ( t ) is given by, ψ ( t ) = 2 ∑ k = 0 L − 1 g k ϕ ( 2 t − k