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

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64CHAPTER 3.HOW TO COUNT THE NUMBER OF SPANNING TREES INGRAPHScolumn. By the matrix-tree theorem, the product n−1 ∏of spanning trees of G.i=1λ i is exactly n times the number□Remark 3.4.7 [21] Let G be an undirected (multi)graph with at least one vertex, andLaplace matrix L with eigenvalues 0 = λ 1 ≥ λ 2 ≥ ... ≥ λ n . Let L ∗ vu be the (v, u)-cofactorof L. Then the number of spanning trees of G equalsτ(G) = L ∗ vu = det(L + 1 n 2 J) = 1 n λ 2...λ n for any v, u ∈ V (G).(The (i, j)-cofactor of a matrix M is by definition (−1) i+j det M(i, j), where M(i, j) isthe matrix obtained from M by deleting row i and column j. Note that L ∗ vu does notdepend on an ordering of the vertices of G)For example, the graph G of valency k on 2 vertices has Laplace matrix L as follows:( ) k −kL =,−k kso that λ 1 = 0, λ 2 = 2k, and τ(G) = 1 2 .2k = k. If we consider the complete graph K n,then λ 2 = ... = λ n = n, and therefore K n has τ(G) = n n−2 spanning trees. This formulawas proposed by Cayley [24]. Remark 4.3.14 is implicit in Kirchhoff [65] and it is knownas the Matrix-Tree Theorem [21].We consider two examples, shown in Figure 3.7 and Figure 3.8.Example 3.4.8 Consider the graph in Figure 3.7Figure 3.7: Example graph, with Laplacian matrix and eigenvalues. Numbers near eachvertex indicate the chosen ordering. The total number of spanning trees can be seen tobe 8 by inspection, which matches with Kirchhoff’s theorem.
3.4. Counting the number of spanning trees in graphs algebraically 65Example 3.4.9 Consider the graph in Figure 3.8Figure 3.8: The 2nd example graph, with Laplacian matrix and eigenvalues. Numbersnear each vertex indicate the chosen ordering. The total number of spanning trees can beseen to be 3 by inspection, which matches with Kirchhoff’s theorem.From Theorem 3.4.6, it is easy to derive the result of Sachs [102], which states that if Gis regular of degree r, thenτ(G) = 1 n−1∏(r − µ i ),ni=1where µ 1 ≤ µ 2 ≤ ... ≤ µ n = r are the eigenvalues of the adjacency matrix A. For example,the Petersen graph in Figure 3.9 is a regular graph. Its adjacency matrix has eigenvalues3, 1, 1, 1, 1, 1, -2, -2, -2, -2, and its Kirchhoff matrix has eigenvalues 0, 2, 2, 2, 2, 2, 5, 5,5, 5. It is concluded that the Petersen graph has 2000 spanning trees [113].Figure 3.9: The Petersen graph.Finally, we mention Temperley’s result [111],τ(G) = 1 n2det(L + J),where J is the n × n matrix all of whose elements are unity.
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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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Page 19 and 20: LIST OF TABLES 17the n-Barrel chain
Page 21: Part IPreliminaries19
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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
Page 85: Part IIIUse of Derived Theoretical
Page 88 and 89: CHAPTER 5.86THE NUMBER OF SPANNING
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CHAPTER 6.114COUNTING THE NUMBER OF
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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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