Thesis Defense of Ji Li

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49 pages, Thesis Defense, Brandeis University, 2007.

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Thesis Defense of Ji Li

  1. 1. Counting Point-Determining Graphs and Prime Graphs Using Joyal’s Theory of Species Dissertation Defense Thesis Advisor: Professor Ira Gessel Ji Li Department of Mathematics Brandeis University 415 South Street, Waltham, MA May 10th, 2007
  2. 2. Theory of Species Point-Determining Graphs Prime Graphs Outline 1 Theory of Species Definition of Species Operations of Species 2 Point-Determining Graphs Point-Determining Graphs Bi-Point-Determining Graphs Point-Determining 2-colored Graphs 3 Prime Graphs Cartesian Product of Graphs Molecular Species and P´lya’s Cycle Index Polynomial o Arithmetic Product of Species Exponentiation Group Exponential Composition of Species J. L. Counting Point-Determining Graphs and Prime Gra
  3. 3. Theory of Species Point-Determining Graphs Prime Graphs Definition of Species Let B be the category of finite sets with bijections. A species (of structures) is a functor F :B→B that generates for each finite set U a finite set F [U ], the set of F -structures on U , and for each bijection σ : U → V a bijection F [σ] : F [U ] → F [V ], which is called the transport of F -structures along σ. Unlabeled F -Structures The symmetric group Sn acts on the set F [n] = F [{1, 2, . . . , n}] by transport of structures. The Sn -orbits under this action are called unlabeled F -structures of order n. J. L. Counting Point-Determining Graphs and Prime Gra
  4. 4. Theory of Species Point-Determining Graphs Prime Graphs Species of Graphs We denote by G the species of (simple) graphs. Then G [U ] is the set of graphs with vertex set U Example 1 3 5 U = {1, 2, 3, 4, 5} 2 4 σ a c e V = {a, b, c, d, e} b d J. L. Counting Point-Determining Graphs and Prime Gra
  5. 5. Theory of Species Point-Determining Graphs Prime Graphs Associated Series of Species Each species F is associated with an exponential generating series xn F (x) = |F [n]| , n! n≥0 a type generating series F (x) = fn xn , n≥0 where fn is the number of unlabeled F -structures of order n, and a cycle index of the species F , denoted ZF , satisfying F (x) = ZF (x, 0, 0, . . . ), F (x) = ZF (x, x2 , x3 , . . . ). J. L. Counting Point-Determining Graphs and Prime Gra
  6. 6. Theory of Species Point-Determining Graphs Prime Graphs Sum of Species An F1 + F2 -structure on a finite set U is either an F1 -structure on U or an F2 -structure on U . = or F1 F2 F1 + F2 Product of Species An F1 F2 -structure on a finite set U is of the form (π; f1 , f2 ), where π is an ordered partition of U with two blocks U1 and U2 , fi is an Fi -structure on Ui for each i. = F1 · F2 F1 F2 J. L. Counting Point-Determining Graphs and Prime Gra
  7. 7. Theory of Species Point-Determining Graphs Prime Graphs Composition of Species An F1 (F2 )-structure on a finite set U is a tuple of the form (π, f, γ), where • π is a partition of U • f is an F1 -structure on the blocks of π • γ is a set of F2 -structures on each block of π. F2 F2 = F2 = F2 F1 ◦ F2 F1 F2 F1 F2 J. L. Counting Point-Determining Graphs and Prime Gra
  8. 8. Theory of Species Point-Determining Graphs Prime Graphs Quotient Species We say that a group A acts naturally on a species F , if for all finite set U , there is an A-action ρU : A × F [U ] → F [U ] so that for each bijection σ : U → V , the following diagram commutes: ρU A × F [U ] − − → F [U ] −−   idA ×F [σ]  F [σ] ρV A × F [V ] − − → F [V ] −− The quotient species of F by A, denoted F/A, is such that for any finite set U , (F/A)[U ] = F [U ]/A. In other words, the set of F/A-structures on U is the set of A-orbits of F -structures on U . J. L. Counting Point-Determining Graphs and Prime Gra
  9. 9. Theory of Species Point-Determining Graphs Prime Graphs Composition with Ek as a Quotient Species Let k be any positive integer. Let Ek be the species of k-element sets. Let F · F · ····F Fk = . k copies We observe that F F F F F F Sk -orbits Ek F k /Sk = Ek ◦ F. J. L. Counting Point-Determining Graphs and Prime Gra
  10. 10. Theory of Species Point-Determining Graphs Prime Graphs Outline 1 Theory of Species Definition of Species Operations of Species 2 Point-Determining Graphs Point-Determining Graphs Bi-Point-Determining Graphs Point-Determining 2-colored Graphs 3 Prime Graphs Cartesian Product of Graphs Molecular Species and P´lya’s Cycle Index Polynomial o Arithmetic Product of Species Exponentiation Group Exponential Composition of Species J. L. Counting Point-Determining Graphs and Prime Gra
  11. 11. Theory of Species Point-Determining Graphs Prime Graphs Neighborhood and Augmented Neighborhood In a graph G, the neighborhood of a vertex v is the set of vertices adjacent to v, the augmented neighborhood of a vertex is the union of the vertex itself and its neighborhood. Example v w1 w2 w3 w4 In the above figure, the neighborhood of the vertex v is the set {w1 , w2 , w3 , w4 }, while the augmented neighborhood of v is the set {v, w1 , w2 , w3 , w4 }. J. L. Counting Point-Determining Graphs and Prime Gra
  12. 12. Theory of Species Point-Determining Graphs Prime Graphs Point-Determining Graphs and Co-Point-Determining Graphs • A graph is called point-determining if no two vertices of this graph have the same neighborhoods. • A graph is called co-point-determining if no two vertices of this graph have the same augmented neighborhoods. Example The graph on the left is co-point-determining, and the graph on the right is point-determining. These two graphs are complements of each other. J. L. Counting Point-Determining Graphs and Prime Gra
  13. 13. Theory of Species Point-Determining Graphs Prime Graphs A Natural Transformation Let P be the species of point-determining graphs, and let Q be the species of co-point-determining graphs. There is a natural transformation α:P →Q that sends each point-determining graph to its complement, which is a co-point-determining graph on the same vertex set, such that the following diagram commutes for any bijection σ : U → V : P[σ] P[U ] − − → P[V ] −−   α  α Q[σ] Q[U ] − − → Q[V ] −− We call the species P isomorphic to the species Q, written as P = Q. J. L. Counting Point-Determining Graphs and Prime Gra
  14. 14. Theory of Species Point-Determining Graphs Prime Graphs Transform a Graph into a Point-Determining Graph 3 3 9 2 9 2 1 5 1 5 8 6 8 6 4 7 4 7 The transformation from a graph G with vertex set [11] = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} to a point-determining graph P with vertex set {{1, 9, 3}, {8}, {4, 7}, {6}, {2, 5}}. J. L. Counting Point-Determining Graphs and Prime Gra
  15. 15. Theory of Species Point-Determining Graphs Prime Graphs Generating Series of Point-Determining Graphs Let E+ be the species of non-empty sets. We get a species identity G = P ◦ E+ , which enables us to enumerate point-determining graphs. For example, we can write down the beginning terms of the exponential generating series and the type generating series of P (previously done by Read): x x2 x3 x4 x5 x6 x7 P(x) = 1+ + +4 +32 +588 +21476 +1551368 +· · · 1! 2! 3! 4! 5! 6! 7! P(x) = 1 + x + x2 + 2 x3 + 5 x4 + 16 x5 + 78 x6 + 588 x7 + · · · J. L. Counting Point-Determining Graphs and Prime Gra
  16. 16. Theory of Species Point-Determining Graphs Prime Graphs Bi-Point-Determining Graphs We denote by R the species of bi-point-determining graphs, which are graphs that are both point-determining and co-point-determining. Example Unlabeled bi-point-determining graphs with no more than 5 vertices. J. L. Counting Point-Determining Graphs and Prime Gra
  17. 17. Theory of Species Point-Determining Graphs Prime Graphs Alternating Phylogenetic Trees A phylogenetic tree is a rooted tree with labeled leaves and unlabeled internal vertices in which no vertex has exactly one child. An alternating phylogenetic tree is either a single vertex, or a phylogenetic tree with more than one labeled vertex whose internal vertices are colored black or white, where no two adjacent vertices are colored the same way. Example 5 4 8 An alternating 6 9 phylogenetic tree on 9 3 vertices, where the root 1 7 is colored black. 2 J. L. Counting Point-Determining Graphs and Prime Gra
  18. 18. Theory of Species Point-Determining Graphs Prime Graphs Transform a Graph into a Bi-Point-Determining Graph 6 3 6 3 8 8 5 5 1 2 4 7 1 2 4 7 6 8 2 5 1 4 On each step, we group vertices with the same neighborhoods or vertices with the same augmented neighborhods. 6 3 Whenever vertices with the same neighborhods are grouped, 8 we connect the corresponding vertices/alternating phylogenetic 5 trees with a black node. 1 2 Whenever vertices with the same augmented neighborhoods 4 7 are grouped, we connected the corresponding vertices/ alternating phylogenetic trees with a white node. 7 6 8 2 5 1 4 Vertices left untouched are not colored. J. L. Counting Point-Determining Graphs and Prime Gra
  19. 19. Theory of Species Point-Determining Graphs Prime Graphs A Species Identity for Bi-Point-Determining Graphs The species of graphs is the composition of the species of bi-point-determining graphs and the species of alternating phylogenetic trees T = T G R T G =R ◦T J. L. Counting Point-Determining Graphs and Prime Gra
  20. 20. Theory of Species Point-Determining Graphs Prime Graphs Generating Series of Bi-Point-Determining Graphs Through calculation, we write functional equations for the exponential generating series and the type generating series of R: x x4 x5 x6 x7 x8 R(x) = +12 +312 +13824 +1147488 +178672128 +· · · 1! 4! 5! 6! 7! 8! R(x) = x + x4 + 6x5 + 36x6 + 324x7 + 5280x8 + · · · J. L. Counting Point-Determining Graphs and Prime Gra
  21. 21. Theory of Species Point-Determining Graphs Prime Graphs Multisort Species Let Bk be the category of finite k-sets with bijective multifunctions. A species of k sorts is a functor F : Bk → B. 2-Colored Graphs A 2-colored graph is a graph in which all vertices are colored either white or black, and no two adjacent vertices are assigned the same color. We denote by G (X, Y ) the 2-sort species of 2-colored graphs, where vertices colored white are of sort X, and vertices colored black are of sort Y . J. L. Counting Point-Determining Graphs and Prime Gra
  22. 22. Theory of Species Point-Determining Graphs Prime Graphs Point-Determining 2-colored Graphs • A 2-colored graph is called point-determining if the underlying graph is point-determining. • A 2-colored graph is called semi-point-determining if all vertices of the same color have distinct neighborhoods. • Note that the graph is semi-point-determining, but it is not point-determining. We denote by • P(X, Y ) — the 2-sort species of point-determining 2-colored graphs • P s (X, Y ) — the 2-sort species of semi-point-determining 2-colored graph • P c (X, Y ) — the 2-sort species of connected point-determining 2-colored graph J. L. Counting Point-Determining Graphs and Prime Gra
  23. 23. Theory of Species Point-Determining Graphs Prime Graphs Functional Equations: Part I The idea is similar to the formula for the species of point-determining graphs P: G = P ◦ E+ . We transform a 2-colored graph into a semi-point-determining 2-colored graph by grouping vertices with the same neighborhoods. Note that if two vertices have the same neighborhoods, then they must be colored in the same way. G (X, Y ) = P s (E+ (X), E+ (Y )). J. L. Counting Point-Determining Graphs and Prime Gra
  24. 24. Theory of Species Point-Determining Graphs Prime Graphs Functional Equations: Part II The observation that a semi-point-determining graph consists of • one or none isolated vertex colored white • one or none isolated vertex colored black • a set (possibly empty) of connected point-determining 2-colored graphs with at least two vertices leads to the functional equation: P s (X, Y ) = (1 + X)(1 + Y ) E (P≥2 (X, Y )) c c c P≥2 (X, Y ) P≥2 (X, Y ) 1+X P s (X, Y ) E 1+Y c c P≥2 (X, Y ) P≥2 (X, Y ) J. L. Counting Point-Determining Graphs and Prime Gra
  25. 25. Theory of Species Point-Determining Graphs Prime Graphs Functional Equations: Part III Similarly, we observe that a point-determining 2-colored graph consists of • one or none isolated vertex, colored white or black • a set of connected point-determining 2-colored graphs with at least two vertices Therefore, c P(X, Y ) = (1 + X + Y ) E (P≥2 (X, Y )) c c P≥2 (X, Y ) P≥2 (X, Y ) P(X, Y ) E 1+X +Y c c P≥2 (X, Y ) P≥2 (X, Y ) J. L. Counting Point-Determining Graphs and Prime Gra
  26. 26. Theory of Species Point-Determining Graphs Prime Graphs Generating Series of Point-Determining 2-colored Graphs These functional equations allow us to calculate the generating series of the species P s (X, Y ), P c (X, Y ), and P(X, Y ). For example, P(x, y) = 1 + x + y+ xy + x2 y + xy 2 + 2x2 y 2 + 3x3 y 2 + 3x2 y 3 + · · · . Unlabeled point-determining 2-colored graphs with no more than 5 vertices. J. L. Counting Point-Determining Graphs and Prime Gra
  27. 27. Theory of Species Point-Determining Graphs Prime Graphs Outline 1 Theory of Species Definition of Species Operations of Species 2 Point-Determining Graphs Point-Determining Graphs Bi-Point-Determining Graphs Point-Determining 2-colored Graphs 3 Prime Graphs Cartesian Product of Graphs Molecular Species and P´lya’s Cycle Index Polynomial o Arithmetic Product of Species Exponentiation Group Exponential Composition of Species J. L. Counting Point-Determining Graphs and Prime Gra
  28. 28. Theory of Species Point-Determining Graphs Prime Graphs Cartesian Product of Graphs The Cartesian product of two graphs G1 and G2 , denoted G1 ⊙ G2 , is the graph whose vertex set is V (G1 ⊙ G2 ) = {(u, v) : u ∈ V (G1 ), v ∈ V (G2 )}, and in which the vertex (u1 , v1 ) is adjacent to the vertex (u2 , v2 ) if either u1 = u2 and v1 is adjacent to v2 or v1 = v2 and u1 is adjacent to u2 . Example 1 1,1’ 2 4 2,1’ 4,1’ 3 3,1’ 1,2’ 2,2’ 4,2’ 1,3’ 1’ 2,3’ 4,3’ 3,2’ 3’ 2’ 3,3’ J. L. Counting Point-Determining Graphs and Prime Gra
  29. 29. Theory of Species Point-Determining Graphs Prime Graphs Properties of the Cartesian Product The Cartesian product is commutative and associative. We write n Gn = ⊡ G. i=1 Prime Graphs A graph G is said to be prime with respect to Cartesian multiplication if G is a non-trivial connected graph such that G ∼ H1 ⊙ H2 implies that either H1 or H2 is a singleton vertex. = Relatively Prime Two graphs G and H are called relatively prime with respect to Cartesian multiplication if and only if G = G1 ⊙ J and H ∼ H1 ⊙ J ∼ = imply that J is a singleton vertex. J. L. Counting Point-Determining Graphs and Prime Gra
  30. 30. Theory of Species Point-Determining Graphs Prime Graphs Decomposition of a Connected Graph Any non-trivial connected graph can be decomposed into prime factors. Sabidussi proved that such a prime factorization is unique up to isomorphism. Example = A connected graph with 24 vertices is decomposed into prime graphs with 2 vertices 3 vertices, and 4 vertices, respectively. J. L. Counting Point-Determining Graphs and Prime Gra
  31. 31. Theory of Species Point-Determining Graphs Prime Graphs Molecular Species • A molecular species is a species that is indecomposable under addition. • If M is molecular, then M = Mn for some n, i.e., M [U ] is nonempty if and only if U is an n-element set. • If M = Mn , then M = X n /A for some subgroup A of Sn . • The X n /A-structures on a finite set U , where |U | = n, is the set of A-orbits of the action A on the set of linear orders on U . In other words, X n /A is the quotient species of X n by A. • Each subgroup A of Sn gives rise to a molecular species X n /A. J. L. Counting Point-Determining Graphs and Prime Gra
  32. 32. Theory of Species Point-Determining Graphs Prime Graphs Cycle Index of a Group • Let A be a subgroup of Sn . The cycle index polynomial of A, defined by P´lya, is o n 1 c (σ) Z(A) = Z(A; p1 , p2 , . . . , pn ) = pkk , |A| σ∈A k=1 where for a permutation σ, ck (σ) is the number of k-cycles in σ. • If a molecular species M = X n /A, then the cycle index of the species M is the same as the cycle index polynomial of the group A. That is, Z(A) = ZX n /A . J. L. Counting Point-Determining Graphs and Prime Gra
  33. 33. Theory of Species Point-Determining Graphs Prime Graphs Species Associated to a Graph Each graph G is associated to a species OG , where the OG -structures on a finite set U is defined to be the set of graphs isomorphic to G with vertex set U . a b b c c d d e e a d c e d a e b a c b e a b c d G OG [{a, b, c, d, e}] OG is Molecular The automorphism group of G acts on the vertex set of G. If G is a graph with n vertices, then aut(G) may be identified with a subgroup of Sn , and Xn OG = . aut(G) J. L. Counting Point-Determining Graphs and Prime Gra
  34. 34. Theory of Species Point-Determining Graphs Prime Graphs Product Group Let A be a subgroup of Sm , and let B be a subgroup of Sn . We define the product group A × B to be the subgroup of Smn such that a) the group operation is (a1 , b1 ) · (a2 , b2 ) = (a1 a2 , b1 b2 ) b b) an element (a, b) of A × B acts on (i, j) for some i ∈ [m] and j ∈ [n] by (a, b)(i, j) = (a(i), b(j)) a J. L. Counting Point-Determining Graphs and Prime Gra
  35. 35. Theory of Species Point-Determining Graphs Prime Graphs Arithmetic Product of Species In the above setting, we start with two molecular species X m /A and X n /B, and get a new molecular species X mn /(A × B), which is defined to be the arithmetic product of X m /A and X n /B: B-orbits Xm Xn X mn ⊡ := . A B A×B A-orbits The arithmetic product of species was previously studied by Maia and M´ndez. e J. L. Counting Point-Determining Graphs and Prime Gra
  36. 36. Theory of Species Point-Determining Graphs Prime Graphs Properties of the Arithmetic Product The arithmetic product has the following properties (given by Maia and M´ndez): e commutativity F1 ⊡ F2 = F2 ⊡ F1 , associativity F1 ⊡ (F2 ⊡ F3 ) = (F1 ⊡ F2 ) ⊡ F3 , distributivity F1 ⊡ (F2 + F3 ) = F1 ⊡ F2 + F1 ⊡ F3 , unit F1 ⊡ X = X ⊡ F1 = F1 . J. L. Counting Point-Determining Graphs and Prime Gra
  37. 37. Theory of Species Point-Determining Graphs Prime Graphs Cartesian Product of Graphs and Arithmetic Product of Species Let G1 and G2 be two graphs that are relatively prime to each other. Then the species associated to the Cartesian product of G1 and G2 is equivalent to the arithmetic product of the species associated to G1 and the species associated to G2 . That is, OG1 ⊙G2 = OG1 ⊡ OG2 Proof Since G1 and G2 are relatively prime, a theorem of Sabidussi gives that aut(G1 ⊙ G2 ) = aut(G1 ) × aut(G2 ). Therefore, X mn X mn OG1 ⊙G2 = = aut(G1 ⊙ G2 ) aut(G1 ) × aut(G2 ) Xm Xn = ⊡ = OG1 ⊡ OG2 . aut(G1 ) aut(G2 ) J. L. Counting Point-Determining Graphs and Prime Gra
  38. 38. Theory of Species Point-Determining Graphs Prime Graphs Exponentiation Group Let A be a subgroup of Sm , and let B be a subgroup of Sn . The exponentiation group B A is a subgroup of Snm , whose group elements are of the form (α, τ ) with α ∈ A and τ : [m] → B. a) The composition of two elements (α, τ ) and α (β, η) is given by α τ (1) (α, τ )(β, η) = (αβ, (τ ◦β)η). τ (5 ) τ (2 ) b) The element (α, τ ) acts on the set of functions from [m] to α [n] by sending each τ( α 4) 3) τ( f : [m] → [n] to g, where for all i ∈ [m], α g(i) = τ (i)(f (α−1 i)). J. L. Counting Point-Determining Graphs and Prime Gra
  39. 39. Theory of Species Point-Determining Graphs Prime Graphs I Operators Let (α, τ ) be an element of the exponentiation group B A such that • α is an m-cycle in the group A • τ = (τ (1), τ (2), . . . , τ (m)) ∈ B m satisfies that the cycle type of τ (m)τ (m − 1) · · · τ (2)τ (1) is λ Palmer and Robinson defined the operators Im on the power sum symmetric functions by Im (pλ ) = pγ , where γ is the cycle type of the element (α, τ ) of B A . More explicitly, γ = (γ1 , γ2 , . . . ) is the partition of nm with gcd(m,l) 1 j cj (γ) = µ ici (λ) . j l l|j i | l/ gcd(m,l) J. L. Counting Point-Determining Graphs and Prime Gra
  40. 40. Theory of Species Point-Determining Graphs Prime Graphs ⊠ Operator • The operation ⊠ on the symmetric functions is defined by letting pν := pλ ⊠ pµ , where ck (ν) = gcd(i, j) ci (λ)cj (µ). lcm(i,j)=k • If a ∈ A has cycle type λ, and b ∈ B has cycle type µ, then (a, b) ∈ A × B has cycle type ν. • If λ = (λ1 , λ2 , . . . ) is a partition of n, then Iλ (pµ ) = Iλ1 (pµ ) ⊠ Iλ2 (pµ ) ⊠ · · · . J. L. Counting Point-Determining Graphs and Prime Gra
  41. 41. Theory of Species Point-Determining Graphs Prime Graphs Cycle Index of Exponentiation Group Theorem (Palmer and Robinson) The cycle index polynomial of B A is the image of Z(B) under the operator obtained by substituting the operator Ir for the variables pr in Z(A). That is, Z(B A ) = Z(A) ∗ Z(B). J. L. Counting Point-Determining Graphs and Prime Gra
  42. 42. Theory of Species Point-Determining Graphs Prime Graphs An Example of the Exponentiation Group Let A = S2 and B = C3 . The element (α, τ ) of B A , with α = (1, 2), τ (1) = id and τ (2) = (1, 2, 3), acts on the set of functions from [2] to [3]. The cycle type of (α, τ ) is (6, 3), which means, I2 (p3 ) = p3 p6 . We can calculate the cycle index of the exponentiation group using Palmer and Robinson’s theorem: 1 9 Z(B A ) = (p + 8p3 + 3p3 p3 + 6p3 p6 ). 18 1 3 2 J. L. Counting Point-Determining Graphs and Prime Gra
  43. 43. Theory of Species Point-Determining Graphs Prime Graphs Exponential Composition of Species We define the molecular B-orbits A- m or species X n /B A to be the or bit s bit s A- exponential composition of X m /A and X n /B: B - or bits B - or bits Xm Xn XN := . A B BA A-or bits A-or bits Or equivalently, B- s ⊡m or bit or Xm Xn Xn bit s B- := A. A B B A-orbits J. L. Counting Point-Determining Graphs and Prime Gra
  44. 44. Theory of Species Point-Determining Graphs Prime Graphs Exponential Composition of a General Species • Recall that the species of k-element sets Ek = X k /Sk . We call Ek F the exponential composition of F of order k. • The cycle index of the exponential composition is ZEk X n /A = Z(Sk ) ∗ Z(A). • Setting E0 F = X, we set E F := Ek F . k≥0 J. L. Counting Point-Determining Graphs and Prime Gra
  45. 45. Theory of Species Point-Determining Graphs Prime Graphs Properties of Exponential Composition The exponential composition of species satisfies the additive properties: k Ek F1 + F2 = Ei F1 ⊡ Ek−i F2 , i=0 E F1 + F2 = E F1 ⊡ E F2 . J. L. Counting Point-Determining Graphs and Prime Gra
  46. 46. Theory of Species Point-Determining Graphs Prime Graphs Prime Power P k Let P be any prime graph, and k any nonnegative integer. Sabidussi showed that the automorphism group of P k is aut(P k ) = aut(P )Sk . Therefore, k k Xk Xn Xn Xn Ek OP = = = = OP k . Sk aut(P ) aut(P )Sk aut(P k ) E OP = X + OP + OP 2 + · · · . J. L. Counting Point-Determining Graphs and Prime Gra
  47. 47. Theory of Species Point-Determining Graphs Prime Graphs Species of Prime Graphs Let G c be the species of connected graphs. Let P be the species of prime graphs. We can write it in terms of the sum of all prime graphs, i.e., P = P OP . We then apply the additive property of the exponential composition: E P =E OP = ⊡ E OP = ⊡(X + OP + OP 2 + · · · ). P P P This means that we get all connected graphs, since each connected graph has a unique prime factorization (Sabidussi)! Therefore, Theorem E P = G c. J. L. Counting Point-Determining Graphs and Prime Gra
  48. 48. Theory of Species Point-Determining Graphs Prime Graphs Cycle Index of the Species of Prime Graphs In order to get a formula for the cycle index of the exponential composition, we generalize Palmer and Robinson’s theorem for the cycle index polynomial of the exponentiation group, and the cycle index of the species of prime graphs can be then calculated, say, using Maple: 1 2 1 2 3 1 ZP = p + p2 + p + p1 p2 + p3 2 1 2 3 1 3 35 4 7 2 7 1 + p + p2 p2 + p1 p3 + p2 + p4 24 1 4 1 3 8 2 4 91 5 19 3 4 2 3 + p + p p2 + p3 p3 + 5p1 p2 + p1 p4 + p2 p3 + p5 15 1 3 1 3 1 2 3 5 + ··· J. L. Counting Point-Determining Graphs and Prime Gra
  49. 49. Theory of Species Point-Determining Graphs Prime Graphs Thank you! J. L. Counting Point-Determining Graphs and Prime Gra

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