}\,\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

journal of medicine. 1995 Dec 14;333(24):1581-7. PubMed PMID: 7477192. Epub 1995/12/14.eng. 6. Hacke W, Kaste M, Bluhmki E, Brozman M, Davalos A, Guidetti D, et al. Thrombolysis with alteplase 3 to 4.5 hours after acute ischemic stroke. The New England journal of medicine. 2008 Sep 25;359(13):1317-29. PubMed PMID: 18815396. Epub 2008/09/26.eng. 7. Sandercock P, Wardlaw JM, Lindley RI, Dennis M, Cohen G, Murray G, et al. The benefits and harms of intravenous thrombolysis with recombinant tissue plasminogen activator within 6 h of acute ischaemic stroke (the third

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

(5):669-72. PubMed PMID: 12875675. Epub 2003/07/24.eng. 14. Yan C, Wang R, Li J, Deng Y, Wu D, Zhang H, et al. HLA-A gene polymorphism defined by high-resolution sequence-based typing in161 Northern Chinese Han people. Genomics Proteomics Bioinformatics. 2003 Nov;1(4):304-9. PubMed PMID: 15629059. Epub2005/01/05. eng. 15. Zhang HW, Tan ZB, Chen RJ, Li JS, Chen G. [Maize starch biosynthesis and its genetic manipulation]. Yi Chuan. 2003 Jul;25(4):455-60. PubMed PMID: 15639908. Epub 2005/01/11. chi. 16. Li PC, Zhang XG, Zhou L, Zeng Y. [Gene optimization is necessary to express HPV

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

References 1. Kohl S, Hamel C. Clinical utility gene card for: Achromatopsia - update 2013. Eur J Hum Genet. 2013 Nov;21(11). PubMed PMID: 23486539; PubMed Central PMCID: PMC3798849. Epub 2013/03/13. 2. Greenberg JP, Sherman J, Zweifel SA, Chen RW, Duncker T, Kohl S, Baumann B, et al. Spectral-domain optical coherence tomography staging and autofluorescence imaging in achromatopsia. JAMA Ophthalmol. 2014 Apr 1; 132(4):437-45. PubMed PMID: 24504161; PubMed Central PMCID: PMC4423754. 3. Pang JJ, Alexander J, Lei B, Deng W, Zhang K, Li Q, Chang B, et al

treatments by Tukey test. Cultivar Treatment (NaCl) 2nd week 3rd week chla in mg/g fresh weight chlb in mg/g fresh weight tot chl carotenoids + xanthophylls in mg/g fresh weight chla in mg/g fresh weight chlb in mg/g fresh weight tot chl carotenoids + xanthophylls in mg/g fresh weight Suba 0 mM 0.60±0.25 a 0.147±0.5 a 0.75±0.29 a 42.8±15.8 a 0.59±0.01 a 0.12±0.01 a 1.18±0.05 a 46.9±0.1 a 50 mM 0.78±0.27 a 0.21±0.08 a 0.98±0.35 a 55.4±15.9 a 1.097±0.15 b 0.17±0.05 b 0.74±0.03 b 58.22±0.56 b 100 mM 1.01±0.07 a 0.255±0.02 a 1.26±0.09 a 65.2±10.4 a 1.06±0.01 c 0.24±0.14 c 1

, 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