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

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6 CONTENTSII Theoretical Part 533 How to count the number of spanning trees in graphs 553.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2 Problematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3 Counting the number of spanning trees in graphs combinatorically . . . . . 563.4 Counting the number of spanning trees in graphs algebraically . . . . . . . 583.4.1 The Matrix-Tree Theorem . . . . . . . . . . . . . . . . . . . . . . . 583.4.2 Other algebraic methods for counting τ(G) . . . . . . . . . . . . . . 633.5 More known results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 New methods to compute the number of spanning trees of Planar Maps 674.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2 Complexity of maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.3.1 Counting the number of spanning trees in a map of type C = C 1 • C 2 684.3.2 Counting the number of spanning trees in a map C . . . . . . . . . 704.3.3 Counting the number of spanning trees in a map of type C= C 1 : C 2 764.3.4 Counting the number of spanning trees in a map of type C= C 1 | C 2 784.3.5 Counting the number of spanning trees in a map of type C= C 1 ‡ C 2 80III Use of Derived Theoretical Results 835 The Number of Spanning Trees of Certain Families of Planar Maps 855.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 The case of one cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.3 The case of two cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.4 The case of n cycles (n ≥ 3) . . . . . . . . . . . . . . . . . . . . . . . . . . 875.5 Particular cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.5.1 The case of k i = k and h i = h (h ≥ 2k + 1) . . . . . . . . . . . . 885.5.2 The case of k i = 1 and h i = h . . . . . . . . . . . . . . . . . . . . . 895.6 Other values of k i = 1 and h i = h . . . . . . . . . . . . . . . . . . . . . . . 905.7 Other uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.7.1 Formula for the Number of Spanning Trees in the n-Home chains . 935.7.2 Formula for the Number of Spanning Trees in The n-Barrel chains . 955.7.3 Formula for the Number of Spanning Trees in the n-Light chains . . 965.8 The number of spanning trees in some particular planar maps . . . . . . . 975.8.1 Formulae for the number of spanning trees in particular planar map 985.8.2 Formulae for the number of spanning trees in the n-Kite chains . . 995.8.3 Formulae for the number of spanning trees in the n-Envelope chains 1025.8.4 Formulae for the number of spanning trees in the n-Diphenylenechains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
CONTENTS 76 Counting the number of spanning trees in the star flower planar map 1076.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.2 The star flower planar map . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.4 An explicit formula for the number of spanning trees in S n,k . . . . . . . . 1107 Maximal Planar Maps 1157.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.2 Calculating the Wiener index in the maximal planar maps . . . . . . . . . 1187.2.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.2.2 Calculation of the Wiener index in the planar maps . . . . . . . . . 1187.3 Formulae for the Number of Spanning Trees in a Maximal Planar Map . . 1237.3.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247.3.2 An explicit formula for the number of spanning trees in E n . . . . . 127Conclusion 133Bibliography 135
Page 3: 1To my daughter Hanin.
Page 6 and 7: 4my wife who has been very supporti
Page 10 and 11: 8 CONTENTS
Page 14 and 15: 12 LIST OF FIGURES7.11 The maps F n
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Page 19 and 20: LIST OF TABLES 17the n-Barrel chain
Page 21: Part IPreliminaries19
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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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CHAPTER 5.86THE NUMBER OF SPANNING
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CHAPTER 6.108COUNTING THE NUMBER OF
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136 BIBLIOGRAPHY[13] N. Biggs, E. L
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138 BIBLIOGRAPHY[46] P. Erdös, and
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140 BIBLIOGRAPHY[77] D. Lotfi, M. E
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142 BIBLIOGRAPHY[107] R. Shrock and
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Auteur :Titre :Directeurs de thèse
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