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

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50 CHAPTER 2. SPANNING TREESIn the following work, we consider only the graphs that do not have loops.Definition 2.4.7 The Laplacian matrix L of an undirected graph G is defined as thedifference between its degree matrix D and its adjacency matrix A.L = D − A.More formally, an undirected graph G = (V G , E G ) of n vertices, weighted by the weightfunction at any edge (v i , v j ) associated weight p(v i , v j ).The Laplacian matrix of G verifies :⎧⎨ deg(v i ) = ∑ nk=1 p(v i, v k ) if i = jL i,j := −p(v i , v j )if i ≠ j and (v i , v j ) ∈ E G⎩0 otherwise.That is, the diagonal elements have values equal to the degree of the corresponding vertices,and the off-diagonal elements are −p(v i , v j ) (the number of edges that connects v iwith v j ) if an edge connects the two vertices, and 0 otherwise.Example 2.4.8 The Laplacian matrix of a labeled graph G shown in Figure 2.7 is givenby:⎛⎞ ⎛⎞ ⎛⎞2 0 0 0 0 1 1 0 2 −1 −1 0L = ⎜ 0 3 0 0⎟⎝ 0 0 4 0 ⎠ − ⎜ 1 0 2 0⎟⎝ 1 2 0 1 ⎠ = ⎜ −1 3 −2 0⎟⎝ −1 −2 4 −1 ⎠0 0 0 1 0 0 1 0 0 0 −1 1Example 2.4.9 The graph G pictured in Figure 2.8 gives:⎡⎤2 −1 0 0 −1 0−1 4 −1 0 −1 −1L =0 −1 3 −1 0 −1⎢ 0 0 −1 2 0 −1⎥⎣−1 −1 0 0 3 −1⎦0 −1 −1 −1 −1 4Remark 2.4.10 Let G be a finite undirected graph without loops. The Laplace matrix ofG is the matrix L indexed by the vertex set of G, with zero row sums, where L vu = −A vufor v ≠ u. If D is the diagonal matrix, indexed by the vertex set of G such that D vv isthe degree (valency) of v, then L = D − A. The matrix Q = D + A is called the signlessLaplace matrix of G. An important property of the Laplace matrix L and the signlessLaplace matrix Q is that they are positive semidefinite. Indeed, one has Q = MM T andL = NN T , if M is the incidence matrix of G and N the directed incidence matrix of thedirected graph obtained by orienting the edges of G in an arbitrary way.
2.4. Matrices associated to a graph 51Depth-First Search AlgorithmGoal: To determine whether or not a given graph G is connected, in which case it producesa spanning tree of G.1. We shall label the vertices of G as 1, 2, . . . based upon the order of traversal asfollow.2. Start with an arbitrary vertex 1.3. Move to any vertex adjacent to the current selection which have not been labeled.If no such vertex exists backtrack to a previously visited vertex.4. Repeat until no more move is possible.5. If all vertices of G are labeled then G is connected and this traversal generates aspanning tree of G.Kirchoff’s AlgorithmGoal: To count the number of spanning trees of a connected labeled graph G.1. Assume that V G = {v 1 , v 2 , ..., v n }.2. Let L be an n × n matrix given by⎧⎨ deg(v i ) if i = j or else(L) ij = −1 if ij ∈ E G⎩0 if ij /∈ E G .3. Compute any cofactor of L. This is the number of spanning trees of G.2.4.5 Notation:1. One more piece of notation is required. If M is a matrix and i is a row-index for Mand j is a column-index for M, let M(i|j) denotes the submatrix of M obtained bydeleting row i and column j from M.We denote by det L(C)[v i ; v j ] = det L(C)(v 1 , ..., ̂v i , ..., ̂v j , ..., v n ).where L(C)[v i ; v j ] is the submatrix, obtained by deleting the v i -th, v j -th rows andthe v i -th, v j -th columns from the Laplacian matrix of C.2. When choosing two vertices from a set of size n, we can pick one and then the otherbut don’t care about the order, so the number of ways is n(n − 1)/2. (The notationfor the number of ways to choose k elements from n elements is ( nk)or Cnk , read "nchoose k". These numbers are called binomial coefficients.
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
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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
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Page 57 and 58: Chapter 3How to count the number of
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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
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Page 81 and 82: 4.3. Main Results 79Theorem 4.3.31
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CHAPTER 5.100THE NUMBER OF SPANNING
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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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