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

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46 CHAPTER 2. SPANNING TREESExample 2.3.4 Below is the kite. To count the spanning trees, observe that four arepaths around the outside cycle in the drawing. The remaining spanning trees use thediagonal edge. Since we must include an edge to each vertex of degree 2, we obtain fourmore spanning trees. The total is eight.Figure 2.6: A graph G and its all spanning treesIn Example 2.3.4, we counted separately the trees that did or did not contain the diagonaledge. This suggests a recursive procedure to count spanning trees. It is clear that thespanning trees of G not containing e are simply the spanning trees of G − e, but how dowe count the trees that contain e? The answer uses an elementary operation on graphs.Remark 2.3.5 We denote by τ(G) the number of spanning trees of a graph G, sometimescalled the complexity of G. If G is not connected then τ(G) = 0, so we can assume thatG is connected from now on. If G ′ is obtained from G by removing all the loops of G,then τ(G ′ ) = τ(G), since a loop can never occur in a spanning tree, i.e., loops don’t affectspanning trees, so we delete them before the computation. Thus, we may assume that Gcontains no loops as well. Multiple edges, however, do remain a possibility.2.4 Matrices associated to a graphHow do we specify a graph? We can list the vertices and edges (with end-points), butthere are other useful representations. Saying that a graph is loopless means that multipleedges are allowed but loops are not. We need to define some matrices as follows.2.4.1 Adjacency MatrixThe adjacency matrix of a graph G is the square matrix A = A(G) indexed by V × V ,which has its entries as: A vv = 0 for all v ∈ V , and if v ≠ u in V then A vu is the numberof edges of G which have vertices v and u at their ends. In other words, the adjacencymatrix A of a graph G, is a (0, 1)-matrix, where 1’s represent adjacent vertices. A has arow for each vertex, and a column for each vertex. If a vertex v is connected to a vertexu, the entry in row v, column u is 1, and so is the entry in row u, column v. For instance,a graph G and its adjacency matrix; see Figure 2.7
2.4. Matrices associated to a graph 47Figure 2.7: A graph G with its adjacency matrix.Remark 2.4.1 An adjacency matrix is determined by a vertex ordering. Every adjacencymatrix is symmetric (a ij = a ji for all i, j). An adjacency matrix of a simple graph G hasentries 0 or 1, with 0s on the diagonal. The degree of v is the sum of the entries in therow for v in either A or M.Example 2.4.2 Consider the labelled graph G in Figure 2.8.Figure 2.8: A graph with 6 vertices.which has the adjacency matrix A2.4.2 Degree Matrix⎡⎤0 1 0 0 1 01 0 1 0 1 1A =0 1 0 1 0 1⎢0 0 1 0 0 1⎥⎣1 1 0 0 0 1⎦0 1 1 1 1 0The degree of a vertex v ∈ V of G denoted by deg G (v); is the number of edges of G whichare incident with v. The degree matrix of G is the diagonal V -by-V matrix D = D(G)such that D vv = deg G (v) for all v ∈ V , and D vu = 0 if v ≠ u; more precisely, the degreematrix D of a graph, is a diagonal matrix with the vertex degrees in the diagonal.
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
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Page 57 and 58: Chapter 3How to count the number of
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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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CHAPTER 5.96THE 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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Auteur :Titre :Directeurs de thèse
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