The critical group of a graph

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1 The critical group of a graph Peter Sin Texas State U., San Marcos, March th, 4.

2 Critical groups of graphs Outline Laplacian matrix of a graph Chip-firing game Smith normal form Some families of graphs with known critical groups Paley graphs Critical group of Paley graphs

3 This talk is about the critical group, a finite abelian group associated with a finite graph.

4 This talk is about the critical group, a finite abelian group associated with a finite graph. The critical group is defined using the Laplacian matrix of the graph.

5 This talk is about the critical group, a finite abelian group associated with a finite graph. The critical group is defined using the Laplacian matrix of the graph. The critical group arises in several contexts;

6 This talk is about the critical group, a finite abelian group associated with a finite graph. The critical group is defined using the Laplacian matrix of the graph. The critical group arises in several contexts; in physics: the Abelian Sandpile model (Bak-Tang-Wiesenfeld, Dhar);

7 This talk is about the critical group, a finite abelian group associated with a finite graph. The critical group is defined using the Laplacian matrix of the graph. The critical group arises in several contexts; in physics: the Abelian Sandpile model (Bak-Tang-Wiesenfeld, Dhar); its combinatorial variant: the Chip-firing game (Björner-Lovasz-Shor, Gabrielov, Biggs);

8 This talk is about the critical group, a finite abelian group associated with a finite graph. The critical group is defined using the Laplacian matrix of the graph. The critical group arises in several contexts; in physics: the Abelian Sandpile model (Bak-Tang-Wiesenfeld, Dhar); its combinatorial variant: the Chip-firing game (Björner-Lovasz-Shor, Gabrielov, Biggs); in arithmetic geometry: Picard group, graph Jacobian (Lorenzini).

9 This talk is about the critical group, a finite abelian group associated with a finite graph. The critical group is defined using the Laplacian matrix of the graph. The critical group arises in several contexts; in physics: the Abelian Sandpile model (Bak-Tang-Wiesenfeld, Dhar); its combinatorial variant: the Chip-firing game (Björner-Lovasz-Shor, Gabrielov, Biggs); in arithmetic geometry: Picard group, graph Jacobian (Lorenzini). We ll consider the problem of computing the critical group for families of graphs.

10 This talk is about the critical group, a finite abelian group associated with a finite graph. The critical group is defined using the Laplacian matrix of the graph. The critical group arises in several contexts; in physics: the Abelian Sandpile model (Bak-Tang-Wiesenfeld, Dhar); its combinatorial variant: the Chip-firing game (Björner-Lovasz-Shor, Gabrielov, Biggs); in arithmetic geometry: Picard group, graph Jacobian (Lorenzini). We ll consider the problem of computing the critical group for families of graphs. The Paley graphs are a very important class of strongly regular graphs arising from finite fields.

11 This talk is about the critical group, a finite abelian group associated with a finite graph. The critical group is defined using the Laplacian matrix of the graph. The critical group arises in several contexts; in physics: the Abelian Sandpile model (Bak-Tang-Wiesenfeld, Dhar); its combinatorial variant: the Chip-firing game (Björner-Lovasz-Shor, Gabrielov, Biggs); in arithmetic geometry: Picard group, graph Jacobian (Lorenzini). We ll consider the problem of computing the critical group for families of graphs. The Paley graphs are a very important class of strongly regular graphs arising from finite fields. We ll say something about the computation of their critical groups, which involves groups, characters and number theory.

12 Critical groups of graphs Outline Laplacian matrix of a graph Chip-firing game Smith normal form Some families of graphs with known critical groups Paley graphs Critical group of Paley graphs

13 Pierre-Simon Laplace (749-87)

14 Γ = (V, E) simple, connected graph.

15 Γ = (V, E) simple, connected graph. L = D A, A adjacency matrix, D degree matrix.

16 Γ = (V, E) simple, connected graph. L = D A, A adjacency matrix, D degree matrix. Think of L as a linear map L : Z V Z V.

17 Γ = (V, E) simple, connected graph. L = D A, A adjacency matrix, D degree matrix. Think of L as a linear map L : Z V Z V. rank(l) = V.

18 Critical group Z V / Im(L) = Z K (Γ)

19 Critical group Z V / Im(L) = Z K (Γ) The finite group K (Γ) is called the critical group of Γ.

20 Critical group Z V / Im(L) = Z K (Γ) The finite group K (Γ) is called the critical group of Γ. Let ε : Z V Z, v V a vv v V a v.

21 Critical group Z V / Im(L) = Z K (Γ) The finite group K (Γ) is called the critical group of Γ. Let ε : Z V Z, v V a vv v V a v. L(ker(ε)) ker(ε), and K (Γ) = Ker(ε)/L(Ker(ε))

22 Kirchhoff s Matrix-Tree Theorem Gustav Kirchhoff (84-887) Kirchhoff s Matrix Tree Theorem For any connected graph Γ, the number of spanning trees is equal to det( L), where L is obtained from L be deleting the row and column corrresponding to any chosen vertex.

Kirchhoff s Matrix-Tree Theorem Gustav Kirchhoff (84-887) Kirchhoff s Matrix Tree Theorem For any connected graph Γ, the number of spanning trees is

23 Kirchhoff s Matrix-Tree Theorem Gustav Kirchhoff (84-887) Kirchhoff s Matrix Tree Theorem For any connected graph Γ, the number of spanning trees is equal to det( L), where L is obtained from L be deleting the row and column corrresponding to any chosen vertex. Also, det( L) = K (Γ) = V V j= λ j.

24 Critical groups of graphs Outline Laplacian matrix of a graph Chip-firing game Smith normal form Some families of graphs with known critical groups Paley graphs Critical group of Paley graphs

25 Rules - 5 A configuration is an assignment of a nonnegative integer s(v) to each round vertex v and v s(v) to the square vertex.

26 Rules - 5 A configuration is an assignment of a nonnegative integer s(v) to each round vertex v and v s(v) to the square vertex. A round vertex v can be fired if it has at least deg(v) chips.

27 Rules A configuration is an assignment of a nonnegative integer s(v) to each round vertex v and v s(v) to the square vertex. A round vertex v can be fired if it has at least deg(v) chips.

28 Rules A configuration is an assignment of a nonnegative integer s(v) to each round vertex v and v s(v) to the square vertex. A round vertex v can be fired if it has at least deg(v) chips.

29 Rules A configuration is an assignment of a nonnegative integer s(v) to each round vertex v and v s(v) to the square vertex. A round vertex v can be fired if it has at least deg(v) chips. The square vertex is fired only when no others can be fired.

30 Rules A configuration is an assignment of a nonnegative integer s(v) to each round vertex v and v s(v) to the square vertex. A round vertex v can be fired if it has at least deg(v) chips. The square vertex is fired only when no others can be fired. A configuration is stable if no round vertex can be fired.

31 Rules A configuration is an assignment of a nonnegative integer s(v) to each round vertex v and v s(v) to the square vertex. A round vertex v can be fired if it has at least deg(v) chips. The square vertex is fired only when no others can be fired. A configuration is stable if no round vertex can be fired. A configuration is recurrent if there is a sequence of firings that lead to the same configuration.

32 Rules A configuration is an assignment of a nonnegative integer s(v) to each round vertex v and v s(v) to the square vertex. A round vertex v can be fired if it has at least deg(v) chips. The square vertex is fired only when no others can be fired. A configuration is stable if no round vertex can be fired. A configuration is recurrent if there is a sequence of firings that lead to the same configuration. A configuration is critical if it is both recurrent and stable.

33 Sample game -

34 Sample game - -4

35 Sample game

36 Sample game

37 Sample game

38 Sample game

39 Sample game

40 Sample game -

41 Sample game - -4

42 Sample game

43 Sample game

44 Sample game

45 Sample game

46 Relation with Laplacian Start with a configuration s and fire vertices in a sequence where each vertex v is fired x(v) times, ending up with configuration s.

47 Relation with Laplacian Start with a configuration s and fire vertices in a sequence where each vertex v is fired x(v) times, ending up with configuration s. s (v) = x(v) deg(v) + (v,w) E x(w)

48 Relation with Laplacian Start with a configuration s and fire vertices in a sequence where each vertex v is fired x(v) times, ending up with configuration s. s (v) = x(v) deg(v) + (v,w) E x(w) s = s Lx

49 Relation with Laplacian Start with a configuration s and fire vertices in a sequence where each vertex v is fired x(v) times, ending up with configuration s. s (v) = x(v) deg(v) + (v,w) E x(w) s = s Lx Theorem Let s be a configuration in the chip-firing game on a connected graph G. Then there is a unique critical configuration which can be reached from s.

50 Relation with Laplacian Start with a configuration s and fire vertices in a sequence where each vertex v is fired x(v) times, ending up with configuration s. s (v) = x(v) deg(v) + (v,w) E x(w) s = s Lx Theorem Let s be a configuration in the chip-firing game on a connected graph G. Then there is a unique critical configuration which can be reached from s. Theorem The set of critical configurations has a natural group operation making it isomorphic to the critical group K (Γ).

51 Critical groups of graphs Outline Laplacian matrix of a graph Chip-firing game Smith normal form Some families of graphs with known critical groups Paley graphs Critical group of Paley graphs

52 Equivalence and Smith normal form Henry John Stephen Smith (86-883) Given an integer matrix X, there exist unimodular integer matrices P and Q such that [ ] Y PXQ =, Y = diag(s, s,... s r ), s s s r.

53 Critical groups of graphs Outline Laplacian matrix of a graph Chip-firing game Smith normal form Some families of graphs with known critical groups Paley graphs Critical group of Paley graphs

54 Trees, K (Γ) = {}.

55 Trees, K (Γ) = {}. Complete graphs, K (K n ) = (Z/nZ) n.

56 Trees, K (Γ) = {}. Complete graphs, K (K n ) = (Z/nZ) n. n-cycle, n 3, K (C n ) = Z/nZ.

57 Trees, K (Γ) = {}. Complete graphs, K (K n ) = (Z/nZ) n. n-cycle, n 3, K (C n ) = Z/nZ. Wheel graphs W n, K (Γ) = (Z/l n ), if n is odd (Biggs). Here l n is a Lucas number.

58 Trees, K (Γ) = {}. Complete graphs, K (K n ) = (Z/nZ) n. n-cycle, n 3, K (C n ) = Z/nZ. Wheel graphs W n, K (Γ) = (Z/l n ), if n is odd (Biggs). Here l n is a Lucas number. Complete multipartite graphs (Jacobson, Niedermaier, Reiner).

59 Trees, K (Γ) = {}. Complete graphs, K (K n ) = (Z/nZ) n. n-cycle, n 3, K (C n ) = Z/nZ. Wheel graphs W n, K (Γ) = (Z/l n ), if n is odd (Biggs). Here l n is a Lucas number. Complete multipartite graphs (Jacobson, Niedermaier, Reiner). Conference graphs on a square-free number of vertices (Lorenzini).

60 Critical groups of graphs Outline Laplacian matrix of a graph Chip-firing game Smith normal form Some families of graphs with known critical groups Paley graphs Critical group of Paley graphs

61 Raymond E. A. C. Paley (97-33)

62 Paley graphs P(q) Vertex set is F q, q = p t (mod 4)

63 Paley graphs P(q) Vertex set is F q, q = p t (mod 4) S = set of nonzero squares in F q

64 Paley graphs P(q) Vertex set is F q, q = p t (mod 4) S = set of nonzero squares in F q two vertices x and y are joined by an edge iff x y S.

65 Some Paley graphs (from Wolfram Mathworld)

66 Paley graphs are Cayley graphs We can view P(q) as a Cayley graph on (Fq, +) with connecting set S Arthur Cayley (8-95)

67 Paley graphs are strongly regular graphs It is well known and easily checked that P(q) is a strongly regular graph and that its eigenvalues are k = q, r = + q and s = q, with multiplicities, q and q, respectively.

68 Critical groups of graphs Outline Laplacian matrix of a graph Chip-firing game Smith normal form Some families of graphs with known critical groups Paley graphs Critical group of Paley graphs

69 David Chandler and Qing Xiang

70 Symmetries K (P(q)) = q where µ = q 4. ( ) q + q k ( ) q q k = q q 3 µ k,

71 Symmetries K (P(q)) = q where µ = q 4. Aut(P(q)) F q S. ( ) q + q k ( ) q q k = q q 3 µ k,

72 Symmetries K (P(q)) = q where µ = q 4. Aut(P(q)) F q S. K (P(q)) = K (P(q)) p K (P(q)) p ( ) q + q k ( ) q q k = q q 3 µ k,

73 Symmetries K (P(q)) = q where µ = q 4. Aut(P(q)) F q S. K (P(q)) = K (P(q)) p K (P(q)) p ( ) q + q k ( ) q q k = q q 3 µ k, Use F q -action to help compute p -part.

74 Symmetries K (P(q)) = q where µ = q 4. Aut(P(q)) F q S. K (P(q)) = K (P(q)) p K (P(q)) p ( ) q + q k ( ) q q k = q q 3 µ k, Use F q -action to help compute p -part. Use S-action to help compute p-part.

75 p -part Joseph Fourier (768-83)

76 Discrete Fourier Transform X, complex character table of (F q, +)

77 Discrete Fourier Transform X, complex character table of (F q, +) X is a matrix over Z[ζ], ζ a complex primitive p-th root of unity.

78 Discrete Fourier Transform X, complex character table of (F q, +) X is a matrix over Z[ζ], ζ a complex primitive p-th root of unity. q XXt = I.

79 Discrete Fourier Transform X, complex character table of (F q, +) X is a matrix over Z[ζ], ζ a complex primitive p-th root of unity. q XXt = I. q XLX t = diag(k ψ(s)) ψ, ()

80 Discrete Fourier Transform X, complex character table of (F q, +) X is a matrix over Z[ζ], ζ a complex primitive p-th root of unity. q XXt = I. q XLX t = diag(k ψ(s)) ψ, () Interpret this as PLQ-equivalence over suitable local rings of integers.

81 Theorem K (P(q)) p = (Z/µZ) µ, where µ = q 4.

82 The p-part Carl Gustav Jacob Jacobi (84-5)

83 F q -action R = Z p [ξ q ], pr maximal ideal of R, R/pR = F q.

84 F q -action R = Z p [ξ q ], pr maximal ideal of R, R/pR = F q. T : F q R Teichmüller character.

85 F q -action R = Z p [ξ q ], pr maximal ideal of R, R/pR = F q. T : F q R Teichmüller character. T generates the cyclic group Hom(F q, R ).

86 F q -action R = Z p [ξ q ], pr maximal ideal of R, R/pR = F q. T : F q R Teichmüller character. T generates the cyclic group Hom(F q, R ). Let R Fq be the free R-module with basis indexed by the elements of F q ; write the basis element corresponding to x F q as [x].

87 F q -action R = Z p [ξ q ], pr maximal ideal of R, R/pR = F q. T : F q R Teichmüller character. T generates the cyclic group Hom(F q, R ). Let R Fq be the free R-module with basis indexed by the elements of F q ; write the basis element corresponding to x F q as [x]. F q acts on R Fq, permuting the basis by field multiplication,

88 F q -action R = Z p [ξ q ], pr maximal ideal of R, R/pR = F q. T : F q R Teichmüller character. T generates the cyclic group Hom(F q, R ). Let R Fq be the free R-module with basis indexed by the elements of F q ; write the basis element corresponding to x F q as [x]. F q acts on R Fq, permuting the basis by field multiplication, R Fq decomposes as the direct sum R[] R F q module with the regular module for F q. of a trivial

89 F q -action R = Z p [ξ q ], pr maximal ideal of R, R/pR = F q. T : F q R Teichmüller character. T generates the cyclic group Hom(F q, R ). Let R Fq be the free R-module with basis indexed by the elements of F q ; write the basis element corresponding to x F q as [x]. F q acts on R Fq, permuting the basis by field multiplication, R Fq decomposes as the direct sum R[] R F q module with the regular module for F q. R F q = q i= E i, E i affording T i. of a trivial

90 F q -action R = Z p [ξ q ], pr maximal ideal of R, R/pR = F q. T : F q R Teichmüller character. T generates the cyclic group Hom(F q, R ). Let R Fq be the free R-module with basis indexed by the elements of F q ; write the basis element corresponding to x F q as [x]. F q acts on R Fq, permuting the basis by field multiplication, R Fq decomposes as the direct sum R[] R F q module with the regular module for F q. R F q = q i= E i, E i affording T i. A basis element for E i is e i = T i (x )[x]. of a trivial x F q

91 S-action Consider action S on R F q. T i = T i+k on S.

92 S-action Consider action S on R F q. T i = T i+k on S. S-isotypic components on R F q are each -dimensional.

93 S-action Consider action S on R F q. T i = T i+k on S. S-isotypic components on R F q are each -dimensional. {e i, e i+k } is basis of M i = E i + E i+k

94 S-action Consider action S on R F q. T i = T i+k on S. S-isotypic components on R F q are each -dimensional. {e i, e i+k } is basis of M i = E i + E i+k The S-fixed subspace M has basis {, [], e k }.

95 S-action Consider action S on R F q. T i = T i+k on S. S-isotypic components on R F q are each -dimensional. {e i, e i+k } is basis of M i = E i + E i+k The S-fixed subspace M has basis {, [], e k }. L is S-equivariant endomorphisms of R Fq, L([x]) = k[x] s S[x + s], x F q.

96 S-action Consider action S on R F q. T i = T i+k on S. S-isotypic components on R F q are each -dimensional. {e i, e i+k } is basis of M i = E i + E i+k The S-fixed subspace M has basis {, [], e k }. L is S-equivariant endomorphisms of R Fq, L([x]) = k[x] s S[x + s], x F q. L maps each M i to itself.

97 Jacobi Sums The Jacobi sum of two nontrivial characters T a and T b is J(T a, T b ) = x F q T a (x)t b ( x).

98 Jacobi Sums The Jacobi sum of two nontrivial characters T a and T b is J(T a, T b ) = x F q T a (x)t b ( x). Lemma Suppose i q and i, k. Then L(e i ) = (qe i J(T i, T k )e i+k )

99 Jacobi Sums The Jacobi sum of two nontrivial characters T a and T b is J(T a, T b ) = x F q T a (x)t b ( x). Lemma Suppose i q and i, k. Then L(e i ) = (qe i J(T i, T k )e i+k ) Lemma (i) L() =. (ii) L(e k ) = ( q([] e k)). (iii) L([]) = (q[] e k ).

100 Corollary The Laplacian matrix L is equivalent over R to the diagonal matrix with diagonal entries J(T i, T k ), for i =,..., q and i k, two s and one zero.

101 Carl Friedrich Gauss ( ) Ludwig Stickelberger (85-936)

102 Gauss and Jacobi Gauss sums: If χ Hom(F q, R ), g(χ) = χ(y)ζ tr(y), y F q where ζ is a primitive p-th root of unity in some extension of R.

103 Gauss and Jacobi Gauss sums: If χ Hom(F q, R ), g(χ) = y F q χ(y)ζ tr(y), where ζ is a primitive p-th root of unity in some extension of R. Lemma If χ and ψ are nontrivial multiplicative characters of F q such that χψ is also nontrivial, then J(χ, ψ) = g(χ)g(ψ). g(χψ)

104 Stickelberger s Congruence Theorem For < a < q, write a p-adically as a = a + a p + + a t p t. Then the number of times that p divides g(t a ) is a + a + + a t.

105 Stickelberger s Congruence Theorem For < a < q, write a p-adically as a = a + a p + + a t p t. Then the number of times that p divides g(t a ) is a + a + + a t. Theorem Let a, b Z/(q )Z, with a, b, a + b (mod q ). Then number of times that p divides J(T a, T b ) is equal to the number of carries in the addition a + b (mod q ) when a and b are written in p-digit form.

106 The Counting Problem k = (q )

107 The Counting Problem k = (q ) What is the number of i, i q, i k such that adding i to q modulo q involves exactly λ carries?

108 The Counting Problem k = (q ) What is the number of i, i q, i k such that adding i to q modulo q involves exactly λ carries? This problem can be solved by applying the transfer matrix method.

109 The Counting Problem k = (q ) What is the number of i, i q, i k such that adding i to q modulo q involves exactly λ carries? This problem can be solved by applying the transfer matrix method. Reformulate as a count of closed walks on a certain directed graph.

110 The Counting Problem k = (q ) What is the number of i, i q, i k such that adding i to q modulo q involves exactly λ carries? This problem can be solved by applying the transfer matrix method. Reformulate as a count of closed walks on a certain directed graph. Transfer matrix method yields the generating function for our counting problem from the adjacency matrix of the digraph.

111 Theorem Let q = p t be a prime power congruent to modulo 4. Then the number of p-adic elementary divisors of L(P(q)) which are equal to p λ, λ < t, is f (t, λ) = min{λ,t λ} i= t t i ( t i i )( ) ( ) t i p + t i ( p) i. λ i The number of p-adic elementary divisors of L(P(q)) which are ) t equal to p t is. ( p+

112 Example:K (P(5 3 )) f (3, ) = 3 3 = 7

113 Example:K (P(5 3 )) f (3, ) = 3 3 = 7 f (3, ) = ( 3 ) ( )( ) 5 3 = 36.

114 Example:K (P(5 3 )) f (3, ) = 3 3 = 7 f (3, ) = ( 3 ) ( )( ) 5 3 = 36. K (P(5 3 )) = (Z/3Z) 6 (Z/5Z) 36 (Z/5Z) 36 (Z/5Z) 5.

115 Example:K (P(5 4 )) f (4, ) = 3 4 = 8.

116 Example:K (P(5 4 )) f (4, ) = 3 4 = 8. f (4, ) = ( 4 ) ( 3 )( ) 5 3 = 44.

117 Example:K (P(5 4 )) f (4, ) = 3 4 = 8. f (4, ) = ( 4 ) ( 3 )( ) 5 3 = 44. f (4, ) = ( 4 ) ( 3 )( ) ( )( ) 5 = 76.

118 Example:K (P(5 4 )) f (4, ) = 3 4 = 8. f (4, ) = ( 4 ) ( 3 )( ) 5 3 = 44. f (4, ) = ( 4 ) ( 3 )( ) ( )( ) 5 = 76. K (P(5 4 )) = (Z/56Z) 3 (Z/5Z) 44 (Z/5Z) 76 (Z/5Z) 44 (Z/65Z) 79.

119 Thank you for your attention!

Smith Normal Forms of Strongly Regular graphs

Smith Normal Forms of Strongly Regular graphs Peter Sin, U. of Florida UD Discrete Math. Seminar, May 7th, 2018. Smith normal forms Smith normal form Smith normal forms associated with graphs Smith and