The critical group of a graph - PDF Free Download

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

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

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

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.

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.

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

Pierre-Simon Laplace (749-87)

Γ = (V, E) simple, connected graph.

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

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.

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.

Critical group Z V / Im(L) = Z K (Γ)

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

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.

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))

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.

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

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.

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.

Rules - - 3 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.

Rules - - 3 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. 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.

Sample game -

Sample game - -4

Sample game 3 - -4-6

Sample game 3 - -4-6 -5 3

Sample game 3 - -4-6 -5 3 3-4

Sample game 3 - -4-6 -5 3 3-4 -4

Sample game

Sample game -

Sample game - -4

Sample game 3 - -4-6 3

Sample game 3 - -4-6 3-5 4

Sample game 3 - -4-6 3-5 -4 4

Sample game 3 - -4-6 3-5 -4-4 4

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.

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)

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.

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

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.

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

Trees, K (Γ) = {}.

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

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

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.

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).

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

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

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

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

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.

Some Paley graphs (from Wolfram Mathworld)

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

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.

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

David Chandler and Qing Xiang

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

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,

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,

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.

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.

p -part Joseph Fourier (768-83)

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

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.

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.

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, ()

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.

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

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

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

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

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 ).

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

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

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.

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

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 }.

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.

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).

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 )

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 ).

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.

Carl Friedrich Gauss (777-855) Ludwig Stickelberger (85-936)

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.

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)

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.

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.

The Counting Problem k = (q )

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?

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.

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.

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+

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

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

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

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

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

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

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

Thank you for your attention!