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

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CHAPTER 5.86THE NUMBER OF SPANNING TREES OF CERTAIN FAMILIES OFPLANAR MAPSFigure 5.1: Example of A map C which has two faces (Cycle)5.3 The case of two cyclesTheorem 5.3.1 Let C be the following map; see Figure 5.2 (It contains two cycles C 1with h 1 edges and C 2 with h 2 edges, and k is the length of the simple path p), thenτ(C) = τ(C 1 ) × τ(C 2 ) − k 2 .Figure 5.2: Case of two cyclesProof:From Theorem 4.3.20, we haveτ(C) = τ(C.p) + kτ(C − p) = (h 1 − k)(h 2 − k) + k[(h 1 − k) + (h 2 − k)]= h 1 h 2 − k 2 = τ(C 1 ) × τ(C 2 ) − k 2 .□Particular case:In the previous Theorem 5.3.1, if k = 1, then τ(C) = τ(C 1 ) × τ(C 2 ) − 1.Example 5.3.2 Consider the following simple map (see Figure. 5.3); using the previousTheorem 5.3.1, it is clear that the number of spanning trees of this map is 23.
5.4. The case of n cycles (n ≥ 3) 87Figure 5.3: A simple planar map has 23 spanning trees5.4 The case of n cycles (n ≥ 3)Let C n be a sequence of maps defined by:• C 1 is a cycle with h 1 edges and it contains the path v 1 , v 2 , ..., v k , v k+1 of length k 1 .• C n is formed by n cycles of length h 1 , h 2 , ..., h n , the i − th cycle and the (i + 1) − thcycle have common path of length k i (see Figure 5.4).Figure 5.4: Case of n cycles
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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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Page 38 and 39: 36 CHAPTER 1. DEFINITIONS AND PROPE
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Page 48 and 49: 46 CHAPTER 2. SPANNING TREESExample
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Page 57 and 58: Chapter 3How to count the number of
Page 59 and 60: 3.3. Counting the number of spannin
Page 69 and 70: Chapter 4New methods to compute the
Page 71 and 72: 4.3. Main Results 69one vertex (any
Page 73 and 74: 4.3. Main Results 71Lemma 4.3.8 Let
Page 75 and 76: 4.3. Main Results 73This recursion
Page 77 and 78: 4.3. Main Results 75Proof: It is th
Page 79 and 80: 4.3. Main Results 77Figure 4.12: An
Page 81 and 82: 4.3. Main Results 79Theorem 4.3.31
Page 83 and 84: 4.3. Main Results 81Property 4.3.33
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Page 110 and 111: 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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