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Counting Prime Graphs and Point-Determining Graphs Using Combinatorial Theory of Species

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Counting Prime Graphs and Point-Determining Graphs Using Combinatorial Theory of Species Counting Prime Graphs and Point-Determining Graphs Using Combinatorial Theory of Species A Dissertation Presented to The Faculty of the Graduate School of Arts and Sciences Brandeis University Department of Mathematics Ira Gessel, Advisor In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by Ji Li August, 2007 The signed version of this signature page is on file at the Graduate School of Arts and Sciences at Brandeis University. This dissertation, directed and approved by Ji Li’s committee, has been accepted and approved by the Faculty of Brandeis University in partial fulfillment of the requirements for the degree of: DOCTOR OF PHILOSOPHY Adam Jaffe, Dean of Arts and Sciences Dissertation Committee: Ira Gessel, Dept. of Mathematics, Chair. Susan Parker, Dept. of Mathematics Alex Postnikov, Dept. of Mathematics, MIT c Copyright by Ji Li 2007 Dedication To Kailang Li and Youzhi Jiang. L`uM`an M`an Q´ıXi¯uYu˘an Xi, W´uJi¯ang Sh`ang Xi`aE´rQi´uSu˘o. by: Q¯uYu´an The way is long with obstacles, I’m questing for the truth to and fro. iv Acknowledgments I would like to thank my advisor, Professor Ira Gessel, for his constant help and support, without which this work would never have existed. I am grateful to the members of my dissertation defense committee, Professor Susan Parker from Brandeis University and Professor Alex Postnikov from MIT. I also owe thanks to the faculty, to my fellow students, and to the kind and supportive staff of the Brandeis Mathematics Department. v Abstract Counting Prime Graphs and Point-Determining Graphs Using Combinatorial Theory of Species A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University, Waltham, Massachusetts by Ji Li In this thesis, we enumerate different types of graphs in light of the combinatorial theory of species initiated by Joyal [13]. The prime graphs with respect to the Cartesian multiplication is enumerated using the exponential composition of species, which is constructed based on the arithmetic product of species studied by Maia and M´endez [18]. The point-determining graphs and bi-point-determining graphs are enumerated by finding functional equations relating them to the species of graphs. vi Contents List of Figures ix Chapter 1. Groups Actions and P´olya’s Cycle Index Polynomial 1 1.1. Symmetric Groups and Group Actions 1 1.2. P´olya’s Cycle Index Polynomials 5 1.3. Exponentiation Group 8 Chapter 2. Combinatorial Theory of Species 15 2.1. Definition of Species 15 2.2. Species Operations 21 2.3. Molecular Species 25 2.4. Compositional Inverse of E+ 31 2.5. Multisort Species 35 2.6. Arithmetic Product of Species 38 Chapter 3. Cartesian Product of Graphs and Prime Graphs 48 3.0. Introduction 48 3.1. Cartesian Product of Graphs 49 3.2. Labeled Prime Graphs 54 3.3. Unlabeled Prime Graphs 61 3.4. Exponential Composition with a Molecular Species 67 3.5. Exponential Composition 78 vii 3.6. Cycle Index of Prime Graphs 82 Chapter 4. Point-Determining Graphs 86 4.0. Introduction 86 4.1. Point-Determining Graphs and Co-Point-Determining Graphs 87 4.2. Connected Point-Determining Graphs and Connected Co-Point- Determining Graphs 91 4.3. Bi-Point-Determining Graphs 96 4.4. 2-Colored Graphs and Connected 2-colored Graphs 112 4.5. Point-Determining 2-Colored Graphs 117 Appendix A. Index of Species 122 Appendix B. General Notations 124 Appendix. Bibliography 127 viii List of Figures 1.1 Group action of A > B. 3 1.2 Group action of A × B. 3 1.3 Group action of B ≀ A. 4 1.4 Group action of BA. 9 1.5 An element of BA. 10 1.6 Cycle type of an element of BA. 10 2.1 Transport of G -structures. 16 2.2 Species associated to a graph. 17 2.3 A linear order and a permutation. 20 2.4 Addition of species. 21 2.5 Multiplication of species. 23 2.6 Composition of species. 23 2.7 The formula (Xm/A) · (Xn/B)= Xm+n/(A × B). 31 2.8 The formula (Xm/A) ◦ (Xn/B)= Xmn/(B ≀ A). 31 2.9 Unlabeled connected graphs. 35 2.10 Multisort species. 37 2.11 The 2-sort species of rooted trees. 38 2.12 A rectangle. 39 ix 2.13 Another rectangle. 39 2.14 A 3-rectangle. 40 2.15 Arithmetic product of species. 42 2.16 Unit of the arithmetic product. 42 2.17 The species E2 ⊡ E2. 43 2.18 The formula (Xm/A) ⊡ (Xn/B)= Xmn/(A × B). 44 2.19 The E2 ⊡ E2-structures. 45 2.20 An element of A × B. 47 3.1 The Cartesian product of graphs. 50 3.2 A prime decomposition. 51 3.3 The automorphism group of a prime power. 56 3.4 The species E2hE2i. 69 3.5 The formula ((Xn/B)⊡m)/A = Xnm /BA. 71 3.6 Unlabeled prime graphs. 84 3.7 Unlabeled non-prime graphs. 85 4.1 Labeled point-determining graphs and co-point-determining graphs. 88 4.2 Transform a graph into a point-determining graph. 89 4.3 Unlabeled point-determining graphs. 91 4.4 Unlabeled connected point-determining graphs and unlabeled connected co-point-determining graphs. 95 4.5 A phylogenetic tree. 96 4.6 A functional equation for the species of phylogenetic trees. 97 x 4.7 Unlabeled phylogenetic trees. 98 E 2 4.8 Unlabeled phylogenetic trees corresponding to the species X 2 . 99 4.9 An alternating phylogenetic tree. 100 4.10 An illustration of (g, h). 101 4.11 Operations OQ and OP . 103 4.12 An illustration of (g′, h′). 103 4.13 Alternating applications of operations OP and OQ. 105 4.14 Construct a graph from a given triple (π,ϕ,γ). 107 4.15 Unlabeled bi-point-determining graphs. 109 4.16 The 2-sort species of 2-colored graphs. 112 4.17 Unlabeled 2-colored graphs counted by the number of edges. 114 4.18 Unlabeled 2-colored graphs. 115 4.19 Unlabeled connected 2-colored graphs. 116 Ps E Pc 4.20 The formula (X,Y )=(1+ X)(1 + Y ) ( ≥2(X,Y )). 118 P E Pc 4.21 The formula (X,Y )=(1+ X + Y ) ( ≥2(X,Y )). 118 4.22 Unlabeled point-determining 2-colored graphs. 120 4.23 Unlabeled point-determining 2-colored graphs with 5 vertices. 121 xi CHAPTER 1 Groups Actions and P´olya’s Cycle Index Polynomial 1.1. Symmetric Groups and Group Actions The symmetric group of order n, denoted Sn, is the group of permutations of [n]= {1, 2,...,n}. Let λ = (λ1,λ2,... ), where the λi are arranged in weakly decreasing order, be a partition of n, denoted λ ⊢ n. That is, |λ| = i λi = n. Let σ be P a permutation of [n]. We say λ is the cycle type of σ, denoted by λ = c. t.(σ), if the λi are the lengths of the cycles in the decomposition of σ into disjoint cycles in c1(λ) c2(λ) weakly decreasing order. Sometimes we write λ = (1 , 2 ,... ), where ck(λ) is the number of parts of length k in λ for k ≥ 1. 1 2 3 4 5 6 Example 1.1.1. Let σ = 4 2 5 1 6 3 = (3, 5, 6)(1, 4)(2). Then the cycle type of σ is (3, 2, 1), and c1(σ)= c2(σ)= c3(σ)=1. It is well-known that the number of permutations of [n] of cycle type λ = (1c1 , 2c2 ,...,kck ) is n! dλ := c c c , c1!1 1 c2!2 2 ··· ck! k k in which the denominator c1 c2 ck zλ := c1!1 c2!2 ··· ck! k 1 CHAPTER 1. GROUP ACTIONS is the number of permutations in Sn that commute with a permutation of cycle type λ. Definition 1.1.2. An action of a group A on a set S is a function ρ : A × S → S, where for x ∈ A and s ∈ S, ρ(x, s) is written x · s. We say this action is natural if both of the following conditions are satisfied: x · (y · s)=(xy) · s, idA ·s = s, for any x, y ∈ A and s ∈ S. Let A be a subgroup of Sm, and let B be a subgroup of Sn. We construct new groups as described below. Definition 1.1.3. We define two groups A > B and A × B both isomorphic to the product of A and B, where A > B is a subgroup of Sm+n and A × B is a subgroup of Smn. The elements of the product groups are of the form (a, b), where a ∈ A and b ∈ B. The group operation is defined by (a1, b1) · (a2, b2)=(a1a2, b1b2), where a1 and a2 are elements of A, and b1 and b2 are elements of B. The group A > B acts on the set [m + n] by a(i), if i ∈{1, 2,...,m}, (1.1.1) (a, b)(i)= b(i − m)+ m, if i ∈{m +1, m +2,...,m + n}. 2 CHAPTER 1. GROUP ACTIONS Figure 1.1 illustrates the action of an element (a, b) of the group A > B. a b Figure 1.1. Group action of A > B. The group A × B acts on the set [m] × [n] and may therefore be viewed as a subgroup of Smn. The action of an element of A × B on an element of [m] × [n] is given by (a, b)(i, j)=(a(i), b(j)), for all i ∈ [m] and j ∈ [n].
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