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

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4my wife who has been very supportive during the period of my research. At last but notleast, I remain unable to conclude without mentioning the kind assistance and thankingall the members of our research group "Laboratory of Mathematics, Computer Scienceand Applications" at the Faculty of sciences Rabat. My thanks of course to all my dears, Icame into contact during these four years and who helped me directly or indirectly duringthe course of this thesis.
ContentsIntroduction 15I Preliminaries 191 Definitions and Properties 211.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.1.1 Vertex Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.1.2 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.2 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.2.1 Preliminaries and notations on maps . . . . . . . . . . . . . . . . . 301.2.2 Faces and Euler’s formula . . . . . . . . . . . . . . . . . . . . . . . 321.2.3 Euler’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.3 Trees and Forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.3.1 Properties of Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.3.2 Spanning Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.4 Distance in trees and graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Spanning trees 412.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.2 Basic concepts and research background . . . . . . . . . . . . . . . . . . . 412.3 Spanning Trees and Enumeration . . . . . . . . . . . . . . . . . . . . . . . 452.3.1 Enumeration of Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 452.3.2 Spanning trees in graphs . . . . . . . . . . . . . . . . . . . . . . . . 452.4 Matrices associated to a graph . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.1 Adjacency Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.2 Degree Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.4.3 Incidence Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.4.4 Laplacian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.4.5 Notation: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
Page 3: 1To my daughter Hanin.
Page 8 and 9: 6 CONTENTSII Theoretical Part 533 H
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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
Page 24 and 25: 22 CHAPTER 1. DEFINITIONS AND PROPE
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Page 46 and 47: 44 CHAPTER 2. SPANNING TREESThe num
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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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CHAPTER 5.86THE NUMBER OF SPANNING
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CHAPTER 6.108COUNTING THE NUMBER OF
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116 CHAPTER 7. MAXIMAL PLANAR MAPSF
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120 CHAPTER 7. MAXIMAL PLANAR MAPSR
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122 CHAPTER 7. MAXIMAL PLANAR MAPSP
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124 CHAPTER 7. MAXIMAL PLANAR MAPS7
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126 CHAPTER 7. MAXIMAL PLANAR MAPSa
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128 CHAPTER 7. MAXIMAL PLANAR MAPST
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130 CHAPTER 7. MAXIMAL PLANAR MAPSP
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134 CHAPTER 7. MAXIMAL PLANAR MAPSI
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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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