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

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120 CHAPTER 7. MAXIMAL PLANAR MAPSRemark 7.2.7 We notice that deleting one edge of multiple edges does not affect theWiener index; through this section, we consider only the simple planar maps, i.e., mapswithout loops and multiple edges.Lemma 7.2.8 Let C n be a simple planar map with n vertices and v be a complete vertexin the map C, thendeg(v) = n − 1 and W (v, C n ) = n − 1.Lemma 7.2.9 Let C n be a simple planar map with n vertices (n ≥ 2) and let v be avertex not complete of C n , thenW (v, C n ) ≥ n.Remark 7.2.101. Let C n be a simple planar map and v be a vertex of C n , then W (v, C n ) ≥ n − 1.2. Let C n be a simple planar map with n vertices, e be an edge of C n and let C n − e bethe map obtained by deleting the edge e such that the map C n −e remains connected,then W (C n − e) ≥ W (C n ).Theorem 7.2.11 Let C n be a simple planar map with n vertices, thenW (C n ) ≥n(n − 1).2Proof: Let C n be a planar map with n vertices and let v be a vertex of C n . By Remark7.2.10, we have: W (v, C n ) ≥ n − 1;W (C n ) = 1 2≥ 1 2≥∑v∈V (C n )∑v∈V (C n )n(n − 1).2W (v, C n )(n − 1) = 1 2 (n − 1) ∑v∈V (C n )Definition 7.2.12 (A maximal planar map [39]) Let E n be a family of planar maps thatcontains:• n vertices, two complete vertices of degree n − 1, two vertices of degree 3 and n − 4vertices of degree 4,• 2(n − 2) faces of degree 3 (all faces having degree 3),• 3(n − 2) edges,1□
7.2. Calculating the Wiener index in the maximal planar maps 121then the family of this maps is called the maximal planar maps if to which no new edgecan be added without violating the planarity of this maps.The maps E 3 and E 4 are presented in the example 7.2.13. For E 5 , E 6 , and E n (see Figure.7.7)Figure 7.7: The maps E 5 , E 6 and E nExample 7.2.13 In the planar maps E 3 and E 4 , all the vertices are completes. In otherwords, we say that the map is complete (see Figure 7.8).Figure 7.8: The maps E 3 and E 4Remark 7.2.14 For all u, v ∈ V (E n ), we have d(u, v) ≤ 2.Proposition 7.2.15 Let E n be the maximal planar map and v be a vertex of E n , thenwe have:1. W (v, E n ) = 2n − deg(v) − 2, 2. W (E n ) = (n − 2) 2 + 2.
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1To my daughter Hanin.
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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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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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