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

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134 CHAPTER 7. MAXIMAL PLANAR MAPSIn future, we plan to work on the planar maps of type C= C 1 ‡ C 2 with ‡ is notat all a simple path, and shall try to generalize the theorem deletion - contraction byworking on deletion more than one edge, i.e., the theorem deletion - contraction workedon one edge but we shall work on n edges and associate it with our results in this thesis.Keywords graphs, maps, trees, spanning trees, complexity, Laplacian matrix, Matrix-Tree Theorem, the n-Fan chains, the n- Grid chains, star flower planar map, maximalplanar map, Wiener index.
Bibliography[1] J. M. Aldous and R. J. Wilson, Graphs and Applications: An Introductory Approach,Undergraduate Mathematics Series, The Open University, Springer-Verlag,2000.[2] G. Al Hagri, Study of some topolgical index of graphs: Wiener and DD(G) index,Ph.D. thesis, Université Mohammed V-Agdal-Faculté des Sciences, Centre d’ÉtudesDoctorales en Sciences et Technologies, 2012.[3] G. Al Hagri, M. Essalih and M. EL marraki, The Wiener Index of SomeParticular Graph, International Journal of Computer Science and Network Security(IJCSNS), Vol. 11 no. 5 (2011), 64 - 69.[4] N. Alon, The number of spanning trees in regular graphs, Random Structures andAlgorithms, 1 (1990), no. 2, 175 - 181.[5] W. N. Anderson & T. D. Morley, Eigenvalues of the Laplacian of a graph, Lin.Multilin. Alg. 18 (1985) 141 - 145.[6] G. Baron, H. Prodinger, R. F. Tichy, F. T. Boesch and J. F. Wang, Thenumber of spanning trees in the square of a cycle, Fibonacci Quarterly 23.3 (1985),258 - 264.[7] S. D. Bedrosian, Generating formulas for the number of trees in a graph, J. FranklinInst, 277 (1964), 313 - 326.[8] S. D. Bedrosian, Formulas for the number of trees in a networks, IEEE Trans.Circuit Theory, CT-8 (1961), 363 - 364.[9] S. D. Bedrosian, Formulas for the number of trees in certain incomplete graphs, J.Franklin Inst, 289 (1970), 67 - 69.[10] C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1973.[11] N. Biggs, Algebraic Graph Theory, London: Cambridge University Press, SecondEdition 1993.[12] N. Biggs, Spanning trees of dual maps, J. Combin. Theory, vol. 11 (1971), 127 -131.135
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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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Page 138 and 139: 136 BIBLIOGRAPHY[13] N. Biggs, E. L
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Page 144 and 145: 142 BIBLIOGRAPHY[107] R. Shrock and
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