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

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128 CHAPTER 7. MAXIMAL PLANAR MAPSTheorem 7.3.3 (A maximal planar map) The complexity of the maximal planar mapE n (see Figure. 7.7) is given by the following formula:τ(E n ) =n ((22 √ + √ 3) n−2 − (2 − √ )3) n−2 , n ≥ 3.3Proof: τ(E 3 ) = 3, τ(E 4 ) = 16, τ(E 5 ) = 75, τ(E n)nhence we obtain the system:⎧⎨⎩τ(E n )n= 4τ(E n−1)n−1τ(E 3 ) = 3τ(E 4 ) = 16= 4τ(E n−1)n−1− τ(E n−2)n−2 , with− τ(E n−2)n−2(by Lemma 7.3.2),Then we get a sequence such that u n = 4u n−1 − u n−2 , n ≥ 3, therefore the characteristicquadratic equation is r 2 − 4r + 1 = 0, so the solutions of this equation are: r 1 = 2 − √ 3and r 2 = 2 + √ 3, hence:τ(E n ) = α(2 − √ 3) n + β(2 + √ 3) n , α, β ∈ R, n ≥ 1.Using the initial conditions τ(E 3)3= 1 and τ(E 4)4= 4, we obtain:α = −2− 7 6√3, β = −2+76√3, hence the result. □Example 7.3.4 The following figure (see Figure. 7.10), illustrates a maximal planar mapwith 4 vertices E 4 and its all spanning trees.Figure 7.10: A maximal planar map E 4 gives rise to 16 spanning trees
7.3. Formulae for the Number of Spanning Trees in a Maximal Planar Map 129Numerical results The Table 7.1 illustrates some of the values of the number of spanningtrees in the maximal planar map E n by using the formula given in Theorem 7.3.3:n 3 4 5 6 7 8 9 10τ(E n ) 3 16 75 336 1463 6240 26199 108640Table 7.1: Some values of τ(E n )Let F n and G n be the maps as follows (see Figure 7.11):Figure 7.11: The maps F n and G nTheorem 7.3.5 (A Fan planar map) The complexity of the Fan planar maps F n and G n(see Figure. 7.11) is given by the following formulae:τ(F n ) = √ 1 ((2 + √ 3) n−1 − (2 − √ 3)), n−1 n ≥ 2,3τ(G n ) = 1 2 (√ 3 − 1)((2 + √ 3) n−1 − (2 − √ )3) n−2 , n ≥ 2.
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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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Page 138 and 139: 136 BIBLIOGRAPHY[13] N. Biggs, E. L
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Page 142 and 143: 140 BIBLIOGRAPHY[77] D. Lotfi, M. E
Page 144 and 145: 142 BIBLIOGRAPHY[107] R. Shrock and
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