American Journal of Numerical Analysis
ISSN (Print): 2372-2118 ISSN (Online): 2372-2126 Website: Editor-in-chief: Emanuele Galligani
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American Journal of Numerical Analysis. 2017, 5(1), 11-15
DOI: 10.12691/ajna-5-1-2
Open AccessArticle

Numerical Solution of Fredholm Integral Equations Using Hosoya Polynomial of Path Graphs

H. S. Ramane1, S.C. Shiralashetti1, R. A. Mundewadi2, and R. B. Jummannaver1

1Department of Mathematics, Karnatak University, Dharwad, India

2P.A. College of Engineering, Mangalore, India

Pub. Date: January 09, 2018

Cite this paper:
H. S. Ramane, S.C. Shiralashetti, R. A. Mundewadi and R. B. Jummannaver. Numerical Solution of Fredholm Integral Equations Using Hosoya Polynomial of Path Graphs. American Journal of Numerical Analysis. 2017; 5(1):11-15. doi: 10.12691/ajna-5-1-2


The main purpose of this paper is to develop the graph theoretic polynomial to solve numerical problems. We present a new method for the solution of Fredholm integral equations using Hosoya polynomials obtained from one of the standard class of graphs called as path. Proposed algorithm expands the desired solution in terms of a set of continuous polynomials over a closed interval [0,1]. However, accuracy and efficiency are dependent on the size of the set of Hosoya polynomials and compared with the existing method.

Fredholm integral equations Hosoya polynomial path

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[1]  L. Caccetta, K. Vijayan, Applications of graph theory, Ars. Combin., 23 (1987), 21-77.
[2]  F. S. Roberts, Graph Theory and Its Applications to the Problems of Society, SIAM Publications, Philadelphia, 1978.
[3]  A.M. Wazwaz. A First Course in Integral Equations. WSPC. New Jersey. 1997.
[4]  U. Lepik, E. Tamme, Application of the Haar wavelets for solution of linearintegral equations, in: Dynamical Systems and Applications, Antala. Proce. (2004). 494-507.
[5]  K. Maleknejad, Y. Mahmoudi, Numerical solution of linear Fredholm integral equation by using hybrid Taylor and Block-Pulse functions, App.Math. Comp. 149 (2004). 799-806.
[6]  K. Maleknejad, F. Mirzaee, Using rationalized Haar wavelet for solvinglinear integral equations, App. Math. Comp. 160 (2005). 579-587.
[7]  K. Maleknejad, M. Yousefi, Numerical solution of the integral equation ofthe second kind by using wavelet bases of hermite cubic splines, App. Math.Comp. 183 (2006). 134-141.
[8]  M. S. Muthuvalu, J. Sulaiman, Half-Sweep Arithmetic Mean method withcomposite trapezoidal scheme for solving linear fredholm integral equations, App. Math. Comp. 217 (2011). 5442-5448.
[9]  F. Harary, Graph Theory, Addison Wesley, Reading, 1968.
[10]  H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc., 69 (1947), 17-20.
[11]  H. Hosoya, On some counting polynomials in chemistry, Discrete Appl.Math., 19 (1988), 239-257.
[12]  E. V. Konstantinova, M. V. Diudea, The Wiener polynomial derivativesand other topological indices in chemical research, Croat. Chem. Acta, 73 (2000), 383-403.
[13]  D. Stevanovic, I. Gutman, I. Hosoya polynomials of trees with up to 11vertices, Zb. Rad. (Kragujevac), 21 (1999), 111-119.
[14]  H. B. Walikar, H. S. Ramane, L. Sindagi, S. S. Shirkol, I. Gutman, Hosoya polynomial of thorn trees, rods, rings and stars, Kragujevac J. Sci., 28(2006), 47-56.
[15]  D. Stevanovic, Hosoya polynomial of composite graphs, Discrete Math., 235 (2001), 237-244.
[16]  I. Gutman, S. Klavzar, M. PetkovÅ¡ek, P. Žigert, On Hosoya polynomials of benzenoid graphs, MATCH Commun. Math. Comput. Chem., 43 (2001), 49-66.
[17]  S. Xu, H. Zhang, The Hosoya polynomial decomposition for cata condensed benzenoid graphs, Discrete Appl. Math., 156 (2008), 2930-2938.
[18]  M. V. Diudea, Hosoya polynomial in tori, MATCH Commun. Math. Comput. Chem., 45 (2002), 109-122.
[19]  S. Xu, H. Zhang, M. V. Diudea, Hosoya polynomials of zig-zag open-ended nanotubes, MATCH Commun. Math. Comput. Chem., 57 (2007), 443-456.
[20]  S. Xu, H. Zhang, Hosoya polynomials of armchair open-ended nanotubes, Int. J. Quantum Chem., 107 (2007), 586-596.
[21]  M. Eliasi, B. Taeri, Hosoya polynomial of zigzag polyhexnanotorus, J.Serb. Chem. Soc., 73 (2008), 311-319.
[22]  S. Klavzar, M. Mollard, Wiener index and Hosoya polynomial of Fibonacci and Lucas cubes, MATCH Commun. Math. Comput. Chem., 68 (2012), 311-324.