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independently, Szekely et al. [ 25 ] arrived at the same idea. If G has k -pendent vertices labeled by v 1 ;v 2 . . .v k , then its terminal distance matrix is the square matrix of order k whose ( i, j )-th entry is d ( v i ,v j \G ). Terminal distance matrices were used for modeling amino acid sequences of proteins and of the genetic code [ 12 , 17 , 18 ]. The terminal Wiener index TW ( G ) of a connected graph G is defined as the sum of the distances between all pairs of its pendent vertices. Thus if V T = { v 1;v 2 ; . . . ,v k } is the set of all pendent

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 the basis of the vertex-degree of graph one of the vertex-degree based index

e-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 1e-03 2.5e-04 1.7500e+02 2.6667e-04 8.3589e-05 2.2476e-05 1e-04 2.5e-05 4.9500e+02 4.1356e-05 1.1371e-05 2.2269e-06 1e-05 2.5e-06 1.7930e+03 6.6194e-06 8.9509e-07 1.7367e-07 1e-06 2.5e-07 5.4810e+03 2.5200e-07 8.5716e-08 2.1467e-08 1e-07 2.5e-08 1.5791e+04 8.1472e-08 1.1234e-08 2.0251e-09 1e-08 2.5e-09 5.6865e+04 2.5045e-09 8.1255e-10 1.5742e-10 1e-09 2.5e-10 1.7420e+05 7.8479e-10 4.0906e-11 2.0754e-11 1e-10 2.5e-11 2.5368e+05 3.2548e-11 4.7299e-13 1.5278e-11

] Chartrand G. and Harary F., Planar permutation graphs , Ann Inst H Poincaré Sec B 3 (1967) 433-438. http://eudml.org/doc/76875 Chartrand G. Harary F. Planar permutation graphs Ann Inst H Poincaré Sec B 3 1967 433 – 438 http://eudml.org/doc/76875 [4] D. Bauer and E. Schmeichel: On a theorem of Häggkvist and Nicoghossian, In Y. Alavi, F.R.K. Chung, R.L. Graham and D. S. Hsu, eds., Graph Theory, Combinatorics, Algorithms, and Applications-Proceedings 2 nd China-USA Graph Theory Conference, SIAM(1991)20-25. Bauer D. Schmeichel E. On a theorem of Häggkvist and Nicoghossian

}\,\sum_{u<v} \left[ d(u,v)^2 + d(uv) \right] \end{equation}$$ the Harary index [ 24 ]: (3) H ( G ) = ∑ u < v 1 d ( u , v ) $$\begin{equation}H(G) = \sum_{u<v} \frac{1}{d(u,v)} \end{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

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 2 $\begin{array}{} \max \left\{ {{d_G}\left( x \right),{d_G}\left( y \right)} \right\} \ge \frac{{n + \alpha }}{2} \end{array}$ for each pair of nonadjacent vertices x and y in G, where α = 3 for odd k and α = 4 for even k

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

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 G and

the First GAMM-Seminar at ICA Stuttgart, October 12-13, Stuttgart, Volume 59 of the series Notes on Numerical Fluid Mechanics (NNFM) 187 202 10.1007/978-3-322-89565-3_17 25 I. S. Kotsireas, (2008), A Survey on Solution Methods for Integral Equations . Available at: http://www.orcca.on.ca/TechReports/TechReports/2008/TR-08-03.pdf Kotsireas I. S. 2008 A Survey on Solution Methods for Integral Equations http://www.orcca.on.ca/TechReports/TechReports/2008/TR-08-03.pdf 26 S. C. Shiralashetti and A. B. Deshi, (2016), An efficient Haar wavelet collocation method for the

, the Haar wavelets method [ 3 ], Legendre wavelets method [ 4 ], Rationalized haar wavelet [ 5 ], Hermite cubic splines [ 6 ], Coifman wavelet scaling functions [ 7 ], CAS wavelets [ 8 ], Bernoulli wavelets [ 9 ], wavelet preconditioned techniques [ 25 , 26 , 27 , 28 ,]. Some of the papers are found for solving Abel′s integral equations using the wavelet based methods, such as Legendre wavelets [ 10 ] and Chebyshev wavelets [ 11 ]. Abel′s integral equations have applications in various fields of science and engineering. Such as microscopy, seismology