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Arithmetic Product of Species

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50 pages, MIT Combinatorics Seminar, 2007.

50 pages, MIT Combinatorics Seminar, 2007.

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  • 1. Cartesian Product of Graphsand Arithmetic Product of Species MIT Combinatorics Seminar Ji Li Department of Mathematics Brandeis University March 9th, 2007
  • 2. Cartesian Product Species Arithmetic Product of Species Exponential Composition of SpeciesWhat This Talk Is About Graphs Species Arithmetic Cartesian Product of Product of Species of Species Graphs Graphs Exponential Composition of Species Prime Graphs J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 3. Cartesian Product Species Definition of Cartesian Product Arithmetic Product of Species Unique Factorization of Connected Graphs Exponential Composition of SpeciesOutline Cartesian Product of Graphs Definition of Cartesian Product Unique Factorization of Connected Graphs Species Definition of Species Species of Graphs Species Associated to A Graph Arithmetic Product of Species Rectangles Arithmetic Product of Species Exponential Composition of Species Cartesian Product and Arithmetic Product Exponential Composition of Species Enumerating Prime Graphs J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 4. Cartesian Product Species Definition of Cartesian Product Arithmetic Product of Species Unique Factorization of Connected Graphs Exponential Composition of SpeciesDefinition of Cartesian Product Definition 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 )}, in which (u, v ) is adjacent to the vertex (w , z) if either u = w and {v , z} ∈ E (G2 ) or v = z and {u, w } ∈ E (G1 ). J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 5. Cartesian Product Species Definition of Cartesian Product Arithmetic Product of Species Unique Factorization of Connected Graphs Exponential Composition of SpeciesAn Example Example 1, 1′ 1, 2′ 1, 3′ 1, 4′ 1 1′ 2′ = 2 3 3′ 4′ 2, 1′ 2, 2′ 3, 1′ 3, 2′ 2, 3′ 2, 4′ 3, 3′ 3, 4′ The Cartesian product of a graph on 3 vertices and a graph on 4 vertices is a graph on 12 vertices. J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 6. Cartesian Product Species Definition of Cartesian Product Arithmetic Product of Species Unique Factorization of Connected Graphs Exponential Composition of SpeciesProperties ◮ commutativity: G1 ⊡ G2 ∼ G2 ⊡ G1 = ◮ associativity: (G1 ⊡ G2 ) ⊡ G3 ∼ G1 ⊡ (G2 ⊡ G3 ) = ◮ · We write ⊡ Gi = G1 ⊡ G2 ⊡ · · · , i ∈I n Gn = ⊡ G. i =1 J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 7. Cartesian Product Species Definition of Cartesian Product Arithmetic Product of Species Unique Factorization of Connected Graphs Exponential Composition of SpeciesPrime Graph Definition A graph G is said to be prime with respect to Cartesian multiplication if G satisfies all of the following conditions: a) G is not a singleton vertex. b) G is connected. c) G ∼ H1 ⊡ H2 implies that H1 or H2 is a singleton vertex. = J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 8. Cartesian Product Species Definition of Cartesian Product Arithmetic Product of Species Unique Factorization of Connected Graphs Exponential Composition of SpeciesRelatively Prime Definition 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 J is a singleton vertex. J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 9. Cartesian Product Species Definition of Cartesian Product Arithmetic Product of Species Unique Factorization of Connected Graphs Exponential Composition of SpeciesDecomposition of a Connected Graph ◮ · If G is a connected graph, then G can be decomposed into prime factors. That is, there is a set {Pi }i ∈I of prime graphs such that G ∼ ⊡ Pi . = i ∈I ◮ · Sabidussi proved that such a prime factorization is unique up to isomorphism. J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 10. Cartesian Product Species Definition of Cartesian Product Arithmetic Product of Species Unique Factorization of Connected Graphs Exponential Composition of SpeciesAn Example Example = ⊡ ⊡ A connected graph on 12 vertices is decomposed into two prime graphs on 2 vertices and one prime graph on 3 vertices. J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 11. Cartesian Product Species Definition of Cartesian Product Arithmetic Product of Species Unique Factorization of Connected Graphs Exponential Composition of SpeciesMonoid Structure of Unlabeled Connected Graphs ◮ · Let M be the set of unlabeled connected graphs. Let P be the set of unlabeled prime graphs. The unique factorization theorem of Sabidussi gives M the structure of a commutative free monoid with a set of primes P. ◮ · This is saying — Every element of M has a unique factorization of the e e e form b11 b22 · · · bkk , where the bi are distinct primes in P. J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 12. Cartesian Product Species Definition of Cartesian Product Arithmetic Product of Species Unique Factorization of Connected Graphs Exponential Composition of SpeciesAn Equation Relating M and P ◮ · Let l (G ), the number of vertices in G , be a length function for M. Then we get an equation relating M and P: 1 1 = . l (G )s 1 − l (P)−s G ∈M P∈P J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 13. Cartesian Product Species Definition of Cartesian Product Arithmetic Product of Species Unique Factorization of Connected Graphs Exponential Composition of SpeciesNumber of Unlabeled Prime Graphs Theorem Let cn be the number of unlabeled connected graphs on n vertices, and let bm be the number of unlabeled prime graphs on m vertices. Then we have cn 1 = . ns (1 − m−s )bm n≥1 m≥2 J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 14. Cartesian Product Species Definition of Cartesian Product Arithmetic Product of Species Unique Factorization of Connected Graphs Exponential Composition of SpeciesA Table ◮ · cn is the number of unlabeled connected graphs on n vertices ◮ · bn is the number of unlabeled prime graphs on n vertices n 1 2 3 4 5 6 7 8 cn 1 1 2 6 21 112 853 11117 bn 1 1 2 5 21 110 853 11111 J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 15. Cartesian Product Species Definition of Cartesian Product Arithmetic Product of Species Unique Factorization of Connected Graphs Exponential Composition of SpeciesUnlabeled Prime Graphs on 4 Vertices Example There are 5 unlabeled prime graphs on 4 vertices. ◮ · In fact, there is only one unlabeled connected graph on 4 vertices that is not prime: J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 16. Cartesian Product Definition of Species Species Species of Graphs Arithmetic Product of Species Species Associated to A Graph Exponential Composition of SpeciesOutline Cartesian Product of Graphs Definition of Cartesian Product Unique Factorization of Connected Graphs Species Definition of Species Species of Graphs Species Associated to A Graph Arithmetic Product of Species Rectangles Arithmetic Product of Species Exponential Composition of Species Cartesian Product and Arithmetic Product Exponential Composition of Species Enumerating Prime Graphs J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 17. Cartesian Product Definition of Species Species Species of Graphs Arithmetic Product of Species Species Associated to A Graph Exponential Composition of SpeciesSpecies Definition A species of structures is a functor from the category of finite sets, with bijections for morphisms, to itself. Example We denote by G the species of graphs. This means the following: ◮ for any finite set U, G [U] is the set of graphs with vertex set U; ◮ any bijection σ:U→V will induce a bijection G [σ] : G [U] → G [V ], J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 18. Cartesian Product Definition of Species Species Species of Graphs Arithmetic Product of Species Species Associated to A Graph Exponential Composition of SpeciesExamples Example ◮ The species of singletons X is defined by setting {U}, if |U| = 1, X [U] = ∅, otherwise. ◮ The species of sets E is defined by setting E [U] = {U}. In other words, the set of E -structures on a given finite set U is a singleton set. ◮ The species of linear orders L . In particular, the species of linear orders on n-element sets is denoted by X n . ◮ The species of connected graphs C . J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 19. Cartesian Product Definition of Species Species Species of Graphs Arithmetic Product of Species Species Associated to A Graph Exponential Composition of SpeciesAddition We can define operations on species, such as = or F1 F2 F1 + F2 Addition: F1 + F2 . J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 20. Cartesian Product Definition of Species Species Species of Graphs Arithmetic Product of Species Species Associated to A Graph Exponential Composition of SpeciesMultiplication = F1 F2 F1 · F2 Multiplication: F1 · F2 . J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 21. Cartesian Product Definition of Species Species Species of Graphs Arithmetic Product of Species Species Associated to A Graph Exponential Composition of SpeciesComposition F2 F2 = F2 = F2 F1 F2 F1 F1 ◦ F2 F2 Substitution: F1 ◦ F2 = F1 (F2 ). J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 22. Cartesian Product Definition of Species Species Species of Graphs Arithmetic Product of Species Species Associated to A Graph Exponential Composition of SpeciesAssociated Series Each species is associated with three counting series— ◮ The exponential generating series F (x) counts labeled F -structures; ◮ The type generating series F (x) counts unlabeled F -structures; ◮ The cycle index ZF is a symmetric function in the variables p1 , p2 , . . . which satisfies F (x) = ZF (x, 0, 0, . . . ), F (x) = ZF (x, x 2 , x 3 , . . . ). J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 23. Cartesian Product Definition of Species Species Species of Graphs Arithmetic Product of Species Species Associated to A Graph Exponential Composition of SpeciesCycle Index of G We can calculate the cycle index of the species G using a formula given by H. D´coste, G. Labelle and P. Leroux: e 2 4 3 2 ZG = p1 + (p1 + p2 ) + p + 2p1 p2 + p3 + 3 1 3 8 4 2 2 4 p1 + 4p1 p2 + 2p2 + p1 p3 + p4 + 3 3 128 5 32 3 2 8 2 4 4 p + p p2 + 8p1 p2 + p1 p3 + p2 p3 + 2p1 p4 + p5 15 1 3 1 3 3 5 + ··· , J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 24. Cartesian Product Definition of Species Species Species of Graphs Arithmetic Product of Species Species Associated to A Graph Exponential Composition of SpeciesConnected Graphs 1 2 8 3 4 5 7 6 The fact that a graph is an assembly of its connected components gives rise to a species identity G = E ◦C, J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 25. Cartesian Product Definition of Species Species Species of Graphs Arithmetic Product of Species Species Associated to A Graph Exponential Composition of SpeciesCycle Index of C ...that can be used to compute the cycle index of connected graphs 1 2 1 1 2 3 ZC = p1 + p + p2 + p3 + p1 + p1 p2 + 2 1 2 3 3 19 4 2 5 2 2 1 p1 + 2p1 p2 + p2 + p1 p3 + p4 + 12 4 3 2 19 3 2 91 5 2 4 2 3 p1 p2 + p2 p3 + p1 + 5p1 p2 + p1 p3 + p5 + p1 p4 3 3 15 3 5 + ··· . J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 26. Cartesian Product Definition of Species Species Species of Graphs Arithmetic Product of Species Species Associated to A Graph Exponential Composition of SpeciesDefinition Definition For any graph G , there is a species associated to G , denoted OG . The OG -structures on a finite set U is the set of graphs isomorphic to G with vertex set U. Example The species E2 is the species of sets with 2 elements. It is also the species associated to the graph J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 27. Cartesian Product Species Rectangles Arithmetic Product of Species Arithmetic Product of Species Exponential Composition of SpeciesOutline Cartesian Product of Graphs Definition of Cartesian Product Unique Factorization of Connected Graphs Species Definition of Species Species of Graphs Species Associated to A Graph Arithmetic Product of Species Rectangles Arithmetic Product of Species Exponential Composition of Species Cartesian Product and Arithmetic Product Exponential Composition of Species Enumerating Prime Graphs J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 28. Cartesian Product Species Rectangles Arithmetic Product of Species Arithmetic Product of Species Exponential Composition of SpeciesDefinition The following definition was given by Maia and Mendez (2006). Definition Let m and n be integers. A rectangle on the set [mn] of height m is a pair (π1 , π2 ) such that a) π1 is a partition of [mn] with m blocks, each of size n b) π2 is a partition of [mn] with n blocks, each of size m c) if B is a block of π1 and B ′ is a block of π2 then |B ∩ B ′ | = 1. ◮ · We denote by N the species of rectangles. J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 29. Cartesian Product Species Rectangles Arithmetic Product of Species Arithmetic Product of Species Exponential Composition of SpeciesA Rectangle Example 12 7 11 6 1 3 4 9 8 10 2 5 ∼ 8 5 2 10 1 9 4 3 12 6 11 7 These two pictures represent the same rectangle. ◮ · This is a rectangle on the set [12] = {1, 2, . . . , 12} of height 3: π1 = {{(1, 3, 4, 9}, {2, 5, 8, 10}, {6, 7, 11, 12}} , π2 = {{1, 8, 12}, {3, 5, 6}, {2, 4, 11}, {7, 9, 10}} . J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 30. Cartesian Product Species Rectangles Arithmetic Product of Species Arithmetic Product of Species Exponential Composition of SpeciesAn Equivalent Picture Example 1 1 A1 9 8 B1 4 12 3 12 7 11 6 8 9 A2 10 2 10 7 B2 5 8 10 2 5 = 12 4 7 2 A3 6 11 11 B3 1 9 4 3 5 3 π1 = {A1 , A2 , A3 } 6 B4 π2 = {B1 , B2 , B3 , B4 } J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 31. Cartesian Product Species Rectangles Arithmetic Product of Species Arithmetic Product of Species Exponential Composition of SpeciesDefinition of k-Rectangle Definition Let m1 , m2 , . . . , mk be integers. Let n = m1 m2 · · · mk . A k-rectangle on the set [n] is a k-tuple of partitions (π1 , π2 , . . . , πk ) such that a) for each i = 1, 2, . . . , k, πi has mi blocks, each of size n/mi . b) for any k-tuple (B1 , B2 , . . . , Bk ), where Bi is a block of πi for each i = 1, 2, . . . , k, we have |B1 ∩ B2 ∩ · · · ∩ Bk | = 1. ◮ · We denote by N (k) the species of k-rectangles. J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 32. Cartesian Product Species Rectangles Arithmetic Product of Species Arithmetic Product of Species Exponential Composition of SpeciesA Picture of a 3-Rectangle A 3-rectangle on [24], labeled on 1111111111 0000000000 A1 A2 A3 A4 triangles. 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 ◮ All triangles are labeled with 1111111111 0000000000 {1, 2, 3, . . . 24}. 1111111111 0000000000 B1 C2 1111111111 0000000000 ◮ π1 has 4 blocks, each of size 6 B2 1111111111 0000000000C 1111111111 0000000000 B3 1 ◮ π2 has 3 blocks, each of size 8 ◮ π3 has 2 blocks, each of size 12 π1 = {A1 , A2 , A3 , A4 } π2 = {B1 , B2 , B3 } π3 = {C1 , C2 } J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 33. Cartesian Product Species Rectangles Arithmetic Product of Species Arithmetic Product of Species Exponential Composition of SpeciesDefinition The following definition was given by Maia and Mendez (2006). Definition Let F1 and F2 be species of structures with F1 [∅] = F2 [∅] = ∅. The arithmetic product of F1 and F2 , denoted F1 ⊡ F2 , is defined as (F1 ⊡ F2 )[U] = F1 [π1 ] × F2 [π2 ], (π1 π2 )∈N [U] where the sum represents the disjoint union and U is a finite set. J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 34. Cartesian Product Species Rectangles Arithmetic Product of Species Arithmetic Product of Species Exponential Composition of SpeciesArithmetic Product In other words, an F1 ⊡ F2 -structure on a finite set U is a tuple of the form ((π1 , f1 ), (π2 , f2 )), where ◮ (π1 , π2 ) is a rectangle on the finite set U ◮ f1 is an F1 -structure on the blocks of π1 ◮ f2 is an F2 -structure on the blocks of π2 J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 35. Cartesian Product Species Rectangles Arithmetic Product of Species Arithmetic Product of Species Exponential Composition of SpeciesA Picture: F1 ⊡ F2 X X X F1 X F2 X X X = X X X X F1 X X X F2 X X J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 36. Cartesian Product Species Rectangles Arithmetic Product of Species Arithmetic Product of Species Exponential Composition of SpeciesProperties of Arithmetic Product The arithmetic product is ◮ commutative: F1 ⊡ F2 = F2 ⊡ F1 ◮ associative: F1 ⊡ (F2 ⊡ F3 ) = (F1 ⊡ F2 ) ⊡ F3 ◮ distributive: F1 ⊡ (F2 + F3 ) = F1 ⊡ F2 + F1 ⊡ F3 ◮ with a unit X : F1 ⊡ X = X ⊡ F1 = F1 J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 37. Cartesian Product Species Rectangles Arithmetic Product of Species Arithmetic Product of Species Exponential Composition of SpeciesAn Illustration of F ⊡ X = F X X F X X = F X X X ◮ · An illustration of the equality F ⊡ X = F. J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 38. Cartesian Product Species Rectangles Arithmetic Product of Species Arithmetic Product of Species Exponential Composition of SpeciesArithmetic Product of {F1 , F2 , . . . , Fk } Definition The arithmetic product of species F1 , F2 , . . . , Fk with Fi (∅) = ∅ for all i is defined by setting k ⊡ Fi = F1 ⊡ F2 ⊡ · · · ⊡ Fk , i =1 which sends each finite set U to the set k ⊡ Fi [U] = F1 [π1 ] × F2 [π2 ] × · · · × Fk [πk ], i =1 (π1 ,π2 ,...,πk )∈N (k) [U] where the sum represents the disjoint union. ◮ · We denote by F ⊡k the arithmetic product of k copies of F . J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 39. Cartesian Product Cartesian Product and Arithmetic Product Species Exponential Composition of Species Arithmetic Product of Species Enumerating Prime Graphs Exponential Composition of SpeciesOutline Cartesian Product of Graphs Definition of Cartesian Product Unique Factorization of Connected Graphs Species Definition of Species Species of Graphs Species Associated to A Graph Arithmetic Product of Species Rectangles Arithmetic Product of Species Exponential Composition of Species Cartesian Product and Arithmetic Product Exponential Composition of Species Enumerating Prime Graphs J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 40. Cartesian Product Cartesian Product and Arithmetic Product Species Exponential Composition of Species Arithmetic Product of Species Enumerating Prime Graphs Exponential Composition of SpeciesThe Arithmetic Product E2 ⊡ X E2 ◮ · The species E2 is the species associated to the graph . ◮ · The species X E2 is the species associated to the graph . ◮ · The species E2 ⊡ X E2 is the species associated to the Cartesian product of and : = J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 41. Cartesian Product Cartesian Product and Arithmetic Product Species Exponential Composition of Species Arithmetic Product of Species Enumerating Prime Graphs Exponential Composition of SpeciesThis is Because.. ◮ · Every automorphism of the product graph is generated by automorphisms of the original graphs: J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 42. Cartesian Product Cartesian Product and Arithmetic Product Species Exponential Composition of Species Arithmetic Product of Species Enumerating Prime Graphs Exponential Composition of SpeciesAutomorphism Group of Two Relatively Prime Graphs ◮ · Sabidussi proved that if G1 and G2 are relatively prime to each other, then aut(G1 ⊡ G2 ) ∼ aut(G1 ) × aut(G2 ). = ◮ · That is, the automorphism group of the product of two relatively prime graphs is the product of the automorphism groups of the graphs. J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 43. Cartesian Product Cartesian Product and Arithmetic Product Species Exponential Composition of Species Arithmetic Product of Species Enumerating Prime Graphs Exponential Composition of SpeciesThe Arithmetic Product E2 ⊡ E2 ◮ · The species E2 is the species associated to the graph . ◮ · The species E2 ⊡ E2 is the species associated to the Cartesian product of and : = J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 44. Cartesian Product Cartesian Product and Arithmetic Product Species Exponential Composition of Species Arithmetic Product of Species Enumerating Prime Graphs Exponential Composition of SpeciesBut.. ◮ · Instead we want the species associated to the graph in which the horizontal and vertical edges are not distinguished. J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 45. Cartesian Product Cartesian Product and Arithmetic Product Species Exponential Composition of Species Arithmetic Product of Species Enumerating Prime Graphs Exponential Composition of SpeciesAutomorphism Group of P k ◮ · Let P be a prime graph. Let k be an integer. ◮ · The automorphism group of P k is, in fact, the so-called exponentiation (Palmer, Robinson) of the symmetric group of order k and the automorphism group of P. ◮ · In particular, k aut(P k ) = aut(P) . J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 46. Cartesian Product Cartesian Product and Arithmetic Product Species Exponential Composition of Species Arithmetic Product of Species Enumerating Prime Graphs Exponential Composition of SpeciesThe species F ⊡k ◮ · Let F be a species of structures with F [∅] = ∅. Let k be a positive integer. An F ⊡k -structure on a finite set U is a tuple of the form ((π1 , f1 ), (π2 , f2 ), . . . , (πk , fk )), where ◮ (π1 , π2 , . . . , πk ) ∈ N (k) [U] is a k-rectangle on U ◮ fi ∈ F [πi ] is an F -structure on the blocks of πi , for each i ∈ [k] J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 47. Cartesian Product Cartesian Product and Arithmetic Product Species Exponential Composition of Species Arithmetic Product of Species Enumerating Prime Graphs Exponential Composition of SpeciesThe Set E2⊡2[4] Example 1 2 1 4 1 4 3 4 3 2 2 3 1 3 1 3 1 2 2 4 4 2 4 3 ⊡2 There are 6 elements in the set E2 [4]. J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 48. Cartesian Product Cartesian Product and Arithmetic Product Species Exponential Composition of Species Arithmetic Product of Species Enumerating Prime Graphs Exponential Composition of SpeciesA Group Action of Sk ◮ · The symmetric group Sk acts on the set F ⊡k [U] by permuting the subscripts of πi and fi , i.e., α((π1 , f1 ), (π2 , f2 ), . . . , (πk , fk )) = ((πα1 , fα1 ), (πα2 , fα2 ), . . . , (παk , fαk )), where ◮ α ∈ Sk is a permutation on [k] ◮ (πα1 , πα2 , . . . , παk ) ∈ N (k) [U] ◮ fαi ∈ F [παi ] for i ∈ [k]. J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 49. Cartesian Product Cartesian Product and Arithmetic Product Species Exponential Composition of Species Arithmetic Product of Species Enumerating Prime Graphs Exponential Composition of SpeciesThe Action of S2 on the Set E2⊡2 [4] ◮ · The Example transposition on the 1 2 1 4 1 4 subscripts of πi 3 4 3 2 2 3 switches rows and columns. ◮ · This action 1 3 1 3 1 2 results in 3 orbits. 2 4 4 2 4 3 J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 50. Cartesian Product Cartesian Product and Arithmetic Product Species Exponential Composition of Species Arithmetic Product of Species Enumerating Prime Graphs Exponential Composition of SpeciesExponential Composition of Species Definition We define the exponential composition of F of order k, denoted Ek F , by setting for each finite set U, (Ek F )[U] be the set of orbits of the action of Sk on F ⊡k [U]. ◮ · We set E0 F = X . Definition We define exponential composition of F , denoted E F , to be the sum of Ek F on all nonnegative integers k, i.e., E F = Ek F . k≥0 J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 51. Cartesian Product Cartesian Product and Arithmetic Product Species Exponential Composition of Species Arithmetic Product of Species Enumerating Prime Graphs Exponential Composition of SpeciesConnected Graphs and Prime Graphs The unique factorization of connected graphs into products of powers of prime graphs leads to the following theorem. Theorem The species C of connected graphs and Π of prime graphs satisfy C =E Π . J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 52. Cartesian Product Cartesian Product and Arithmetic Product Species Exponential Composition of Species Arithmetic Product of Species Enumerating Prime Graphs Exponential Composition of SpeciesEnumeration Theorem ◮ · Palmer and Robinson (1973) proved a theorem which is useful for computing the cycle index of the species of prime graphs. Let A and B be permutation groups acting on [m] and [n], respectively. Then the cycle index polynomial of the exponentiation of A and B is the image of the cycle index polynomial of B under the operator obtained by substituting a certain kind of operators indexed by r for the variables pr in the cycle index polynomial of A. J. L. Cartesian Product of Graphs and Arithmetic Product of Species
  • 53. Cartesian Product Cartesian Product and Arithmetic Product Species Exponential Composition of Species Arithmetic Product of Species Enumerating Prime Graphs Exponential Composition of SpeciesThe End Thank you. J. L. Cartesian Product of Graphs and Arithmetic Product of Species