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

More documents

Recommendations

Info

62CHAPTER 3.HOW TO COUNT THE NUMBER OF SPANNING TREES INGRAPHS⎛D =⎜⎝2 0 0 0 0 0 0 00 2 0 0 0 0 0 00 0 3 0 0 0 0 00 0 0 3 0 0 0 00 0 0 0 3 0 0 00 0 0 0 0 3 0 00 0 0 0 0 0 2 00 0 0 0 0 0 0 2⎞⎛, A =⎟ ⎜⎠ ⎝Then the matrix L = D − A is the 8 × 8 matrix⎛L =⎜⎝2 −1 0 −1 0 0 0 0−1 2 −1 0 0 0 0 00 −1 3 −1 0 −1 0 0−1 0 −1 3 −1 0 0 00 0 0 −1 3 −1 0 −10 0 −1 0 −1 3 −1 00 0 0 0 0 −1 2 −10 0 0 0 −1 0 −1 20 1 0 1 0 0 0 01 0 1 0 0 0 0 00 1 0 1 0 1 0 01 0 1 0 1 0 0 00 0 0 1 0 1 0 10 0 1 0 1 0 1 00 0 0 0 0 1 0 10 0 0 0 1 0 1 0Let now s = 3. By deleting the s-th row and s-th column of L we have the 7 × 7 matrix⎛L ∗ =⎜⎝2 −1 −1 0 0 0 0−1 2 0 0 0 0 0−1 0 3 −1 0 0 00 0 −1 3 −1 0 −10 0 0 −1 3 −1 00 0 0 0 −1 2 −10 0 0 −1 0 −1 2The determinant of this matrix is 56. Thus the Matrix Tree Theorem states that thenumber of spanning trees for this graph is 56. These computations were performed byusing Maple.Example 3.4.5 In the labelled graph G shown in Figure 2.8, by using the matrix-treetheorem , we find a cofactor by removing the last row and column, and then taking thedeterminant.2 −1 0 0 −1−1 4 −1 0 −1τ(G) =0 −1 3 −1 0= 55.0 0 −1 2 0∣−1 −1 0 0 3 ∣⎞⎟⎠⎞⎟⎠⎞⎟⎠
3.4. Counting the number of spanning trees in graphs algebraically 633.4.2 Other algebraic methods for counting τ(G)Some other methods for counting τ(G) are as follows. Let A be an n × n matrix. Theeigenvalues of A are defined as the n roots of the characteristic polynomial det(λI − A).The spectrum of A consists of all eigenvalues of A. All eigenvalues of a real symmetricmatrix are real. For further detail and more explanation on properties and significance ofthe theory of spectra, we refer the reader to [5, 11, 21, 35, 57, 52, 53, 92, 80, 81, 83, 91, 93].The Laplacian matrix of a graph and its eigenvalues can be used in several areas ofmathematical research and have a physical interpretation in various physical andchemical theories. The related matrix: the adjacency matrix of a graph and its eigenvalueswere extensively investigated in the past than the Laplacian matrix. From theLaplace spectrum of a graph, one can determine the number of spanning trees (whichwill be nonzero only if the graph is connected). Furthermore, by basic knowledge oflinear algebra, all eigenvalues of the Kirchhoff matrix L of a graph with n verticesare non-negative, and 0 is one of its eigenvalues because all its row vectors add up tothe 0 vector. So we can let λ 1 ≥ λ 2 ≥ ... ≥ λ n (= 0) denote all eigenvalues of L.Kel’mans and Chelnokov [64] have shown that the number of distinct spanning treesof a graph G is equal to the product of nonzero eigenvalues of the combinatorial Laplacian.For a graph G, the combinatorial Laplacian has non-negative eigenvalues exceptone, λ 1 ≥ λ 2 ≥ ... ≥ λ n (= 0). The number of spanning trees, can be related to theeigenvalues of L as follows: (A proof can be found in [11]. For completeness, we brieflydescribe the proof here). We now can state the following theorem and its proof.Theorem 3.4.6 (Kirchhoff’s Matrix Tree Theorem) For a given undirected connectedgraph G with n vertices, let λ 1 , . . . , λ n−1 be the non-zero eigenvalues of L(G). Then,the number of distinct spanning trees τ(G) is equal to:τ(G) = 1 n−1∏λ i .nEquivalently, τ(G) is equal to the absolute value of any cofactor of the Laplacian matrixof G.Proof: Suppose we consider the characteristic polynomial p(x) of the combinatorialLaplacian L.p(x) = det(L − xI).The coefficient of the linear term is exactlyn−1−i=1∏λ i .i=1On the other hand, the coefficient of the linear term of p(x) is -1 times the sum of thedeterminant of n principal submatrix of L obtained by deleting the i-th row and i-th
Page 3:
1To my daughter Hanin.
Page 6 and 7:
4my wife who has been very supporti
Page 8 and 9:
6 CONTENTSII Theoretical Part 533 H
Page 10 and 11:
8 CONTENTS
Page 14 and 15: 12 LIST OF FIGURES7.11 The maps F n
Page 16: 14 LIST OF TABLES
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
Page 44 and 45: 42 CHAPTER 2. SPANNING TREESFigure
Page 46 and 47: 44 CHAPTER 2. SPANNING TREESThe num
Page 48 and 49: 46 CHAPTER 2. SPANNING TREESExample
Page 50 and 51: 48 CHAPTER 2. SPANNING TREESFor exa
Page 52 and 53: 50 CHAPTER 2. SPANNING TREESIn the
Page 54 and 55: 52 CHAPTER 2. SPANNING TREES
Page 57 and 58: Chapter 3How to count the number of
Page 59 and 60: 3.3. Counting the number of spannin
Page 63: 3.4. 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
Page 85: Part IIIUse of Derived Theoretical
Page 88 and 89: CHAPTER 5.86THE NUMBER OF SPANNING
Page 110 and 111: CHAPTER 6.108COUNTING THE NUMBER OF
Page 112 and 113: CHAPTER 6.110COUNTING THE NUMBER OF
Page 114 and 115:
CHAPTER 6.112COUNTING THE NUMBER OF
Page 116 and 117:
CHAPTER 6.114COUNTING THE NUMBER OF
Page 118 and 119:
116 CHAPTER 7. MAXIMAL PLANAR MAPSF
Page 120 and 121:
Page 122 and 123:
120 CHAPTER 7. MAXIMAL PLANAR MAPSR
Page 124 and 125:
122 CHAPTER 7. MAXIMAL PLANAR MAPSP
Page 126 and 127:
124 CHAPTER 7. MAXIMAL PLANAR MAPS7
Page 128 and 129:
126 CHAPTER 7. MAXIMAL PLANAR MAPSa
Page 130 and 131:
128 CHAPTER 7. MAXIMAL PLANAR MAPST
Page 132 and 133:
130 CHAPTER 7. MAXIMAL PLANAR MAPSP
Page 134 and 135:
Page 136 and 137:
134 CHAPTER 7. MAXIMAL PLANAR MAPSI
Page 138 and 139:
136 BIBLIOGRAPHY[13] N. Biggs, E. L
Page 140 and 141:
138 BIBLIOGRAPHY[46] P. Erdös, and
Page 142 and 143:
140 BIBLIOGRAPHY[77] D. Lotfi, M. E
Page 144 and 145:
142 BIBLIOGRAPHY[107] R. Shrock and
Page 146 and 147:
Auteur :Titre :Directeurs de thèse
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?