enumeration of the number of spanning trees in some ... - Toubkal

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136 BIBLIOGRAPHY[13] N. Biggs, E. Lloyd and R. Wilson, Graph Theory, 1736 - 1936, Clarendon Press,Oxford 1976.[14] F. T. Boesch, H. Prodinger, Spanning Trees Formulas and Chebyshev Polynomials,Graph and Combinatorics, 2 (1986), 191 - 200.[15] F. T. Boesch, J. F. Wang, A Conjecture on the Number of Spanning Trees in theSquare of a Cycle, In: Notes from New York Graph Theory Day V, P. 16, New York:New York Academy Sciences, 1982.[16] Z. R. Bogdanowicz, Formulas for the Number of Spanning Trees in a Fan , AppliedMathematical Sciences, Vol. 2, 2008, no. 16, 781 - 786.[17] B. Bollobás, Modern Graph Theory, Springer, 2002.[18] B. Bollobás, Extremal Graph Theory, Academic Press Inc., London, 1978.[19] J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, 2008.[20] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, AmericanElsevier, New York, 1976.[21] A. E. Brouwer and W. H. Haemers, Spectra of graphs, Springer, 2011[22] T.J.N. Brown, R.B. Mallion, P. Pollak, and A. Roth, Some methods forcounting the spanning trees in labelled molecular graphs, examined in relation tocertain fullerenes, Discrete Applied Mathematics, 67 (1996), no. 1-3, 51 - 66.[23] F. Buckley, F. Harary, Distance in Graphs, Addison-Wesley, Reading, MA, 1990.[24] G. A. Cayley, A Theorem on Trees. Quart. J. Math., 23, (1889) 276-378.[25] A. Cayley, On the theory of analytical forms called trees, Philadelphia Magazine,13 (1857), 172 - 176.[26] G. Chartrand, L. Lesniak, Graphs and digraphs, 3rd edition, Chapman & Hall,1996.[27] T. Chow, The Q-Spectrum and Spanning Trees of Tensor Products of BipartiteGraphs, Proc. Amer. Math. Soc. 125 (1997), 3155 - 3161.[28] F.R.K. Chung, Diameters of Communication Networks, Mathematics Subject classification,Bell Communications Research, Morristown, 1980, New Jersey.[29] F. R. K. Chung and R. P. Langlands, A combinatorial Laplacian with vertexweights, J. Comb. Theory, (A), 75 (1996), 316 - 327.[30] K. L. Chung, W. M. Yan, On the number of spanning trees of a multi-complete/starrelated graph, Information Processing Letters, 76 (2000), 113 - 119.
BIBLIOGRAPHY 137[31] F. R. K. Chung and S. -T. Yau, A combinatorial trace formula, Tsing Hua Lectureson Geometry and Analysis, (ed. S.-T. Yau), International Press, Cambridge,Massachusetts, 1997, 107 - 116.[32] F. Chung and S.-T. Yau, Coverings, heat kernels and spanning trees, ElectronicJournal of Combinatorics 6 (1999), 12 - 21.[33] F. R. K. Chung and S. -T. Yau, Eigenvalue inequalities of graphs and convexsubgraphs, Communications on Analysis and Geometry, 5 (1998), 575 - 624.[34] C. J. Colbourn, The combinatorics of network reliability, International Series ofMonographs on Computer Science, The Clarendon Press Oxford University Press,New York, 1987.[35] D. Cvetkovič, M. Doob, I. Gutman and Torĝasev, Recent Results in theTheory of Graph Spectra, Annals of Discrete Mathematics 36, North-Holland, Amsterdam,1988.[36] D. Cvetkovič, M. Doob and H. Sachs, Spectra of Graphs: Theory and Applications,Third Edition, Johann Ambrosius Barth, Heidelberg, 1995.[37] D. Cvetkovič and I. Gutman, A new method for determining the number ofspanning trees, Publ. Inst. Math. (Beograd) 29 (1981), 49 - 52.[38] M. Desjarlais and R. Molina, Counting spanning trees in grid graphs, CongressusNumerantium, 145: 177 - 185, 2000.[39] R. Diestel, Graph Theory. Electronic Edition, Springer-Verlag New York, 2000.[40] A. A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theoryand applications, Acta Appl. Math. 66 (2001), no. 3, 211 - 249.[41] O. Eǧecioǧlu, J. B. Remmel, A bijections for spanning trees of complete multipartitegraphs, Congr. Numer., 100 (1994), 225 - 243.[42] O. Eǧecioǧlu, J. B. Remmel, Bijections for Cayley trees, spanning trees and theirq-analogues, J. Combin. Theory, (A) 42 (1986), 15 - 30.[43] M. El Marraki, Compositions des fonctions rationnelles et ses aspects combinatoirese,Ph.D. thesis, Université Bordeaux, 2001.[44] M. El Marraki, G. Al Hagri, Calculation of the Wiener index for some particulartrees, Journal of Theoretical and Applied Information Technology (JATIT), Vol. 22,2010, no. 2, 77 - 83.[45] M. El Marraki, A. Modabish and G. Al Hagri, The distance between twovertices of a tree, to be published.
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1To my daughter Hanin.
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4my wife who has been very supporti
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6 CONTENTSII Theoretical Part 533 H
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8 CONTENTS
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12 LIST OF FIGURES7.11 The maps F n
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14 LIST OF TABLES
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LIST OF TABLES 17the n-Barrel chain
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Part IPreliminaries19
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22 CHAPTER 1. DEFINITIONS AND PROPE
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42 CHAPTER 2. SPANNING TREESFigure
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44 CHAPTER 2. SPANNING TREESThe num
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46 CHAPTER 2. SPANNING TREESExample
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48 CHAPTER 2. SPANNING TREESFor exa
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50 CHAPTER 2. SPANNING TREESIn the
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52 CHAPTER 2. SPANNING TREES
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Chapter 3How to count the number of
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3.3. Counting the number of spannin
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Chapter 4New methods to compute the
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4.3. Main Results 69one vertex (any
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4.3. Main Results 71Lemma 4.3.8 Let
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4.3. Main Results 73This recursion
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4.3. Main Results 75Proof: It is th
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4.3. Main Results 77Figure 4.12: An
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4.3. Main Results 79Theorem 4.3.31
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4.3. Main Results 81Property 4.3.33
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Part IIIUse of Derived Theoretical
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Page 140 and 141: 138 BIBLIOGRAPHY[46] P. Erdös, and
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