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Integer Invariants of Abelian Cayley Graphs Deelan Jalil James Madison University July 26, 2013 Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 1 / 29 Cube example. . Deelan Jalil (2013) Integer Invariants of

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Integer Invariants of Abelian Cayley Graphs

Deelan Jalil

James Madison University

July 26, 2013

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 1 / 29

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Cube example.

.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 2 / 29

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Cube example.

            (0, 0, 0) (0, 0, 1) (0, 1, 0) (0, 1, 1) (1, 0, 0) (1, 0, 1) (1, 1, 0) (1, 1, 1) (0, 0, 0) (0, 0, 1) (0, 1, 0) (0, 1, 1) (1, 0, 0) (1, 0, 1) (1, 1, 0) (1, 1, 1)            

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 3 / 29

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SLIDE 4

Cube example.

            (0, 0, 0) (0, 0, 1) (0, 1, 0) (0, 1, 1) (1, 0, 0) (1, 0, 1) (1, 1, 0) (1, 1, 1) (0, 0, 0) 1 1 1 (0, 0, 1) 1 1 1 (0, 1, 0) 1 1 1 (0, 1, 1) 1 1 1 (1, 0, 0) 1 1 1 (1, 0, 1) 1 1 1 (1, 1, 0) 1 1 1 (1, 1, 1) 1 1 1            

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 4 / 29

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SLIDE 5

Cube example.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 5 / 29

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Cube example.

            1 1 1 1 1 1 3 3            

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 6 / 29

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SLIDE 7

Smith normal form.

Given any integer matrix A, we can perform row and column

  • perations so that:

A =        d1 · · · d2 · · · d3 · · · . . . . . . . . . ... . . . · · · dn        where d1|d2 , d2|d3, ... ,dn−1|dn.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 7 / 29

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Smith normal form.

Given any integer matrix A, we can perform row and column

  • perations so that:

A =        d1 · · · d2 · · · d3 · · · . . . . . . . . . ... . . . · · · dn        where d1|d2 , d2|d3, ... ,dn−1|dn. Swap any two rows/columns.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 7 / 29

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SLIDE 9

Smith normal form.

Given any integer matrix A, we can perform row and column

  • perations so that:

A =        d1 · · · d2 · · · d3 · · · . . . . . . . . . ... . . . · · · dn        where d1|d2 , d2|d3, ... ,dn−1|dn. Swap any two rows/columns. Multiply any row/column by a nonzero integer.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 7 / 29

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Smith normal form.

Given any integer matrix A, we can perform row and column

  • perations so that:

A =        d1 · · · d2 · · · d3 · · · . . . . . . . . . ... . . . · · · dn        where d1|d2 , d2|d3, ... ,dn−1|dn. Swap any two rows/columns. Multiply any row/column by a nonzero integer. Add a integer multiple of one row/column to another.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 7 / 29

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Cube example part 2.

            (0, 0, 0) (0, 0, 1) (0, 1, 0) (0, 1, 1) (1, 0, 0) (1, 0, 1) (1, 1, 0) (1, 1, 1) (0, 0, 0) 1 1 1 (0, 0, 1) 1 1 1 (0, 1, 0) 1 1 1 (0, 1, 1) 1 1 1 (1, 0, 0) 1 1 1 (1, 0, 1) 1 1 1 (1, 1, 0) 1 1 1 (1, 1, 1) 1 1 1            

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 8 / 29

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Cube example part 2.

            (0, 0, 0) (0, 0, 1) (0, 1, 0) (0, 1, 1) (1, 0, 0) (1, 0, 1) (1, 1, 0) (1, 1, 1) (0, 0, 0) −1 −1 −1 (0, 0, 1) −1 −1 −1 (0, 1, 0) −1 −1 −1 (0, 1, 1) −1 −1 −1 (1, 0, 0) −1 −1 −1 (1, 0, 1) −1 −1 −1 (1, 1, 0) −1 −1 −1 (1, 1, 1) −1 −1 −1            

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 9 / 29

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Cube example part 2.

            (0, 0, 0) (0, 0, 1) (0, 1, 0) (0, 1, 1) (1, 0, 0) (1, 0, 1) (1, 1, 0) (1, 1, 1) (0, 0, 0) 3 −1 −1 −1 (0, 0, 1) −1 3 −1 −1 (0, 1, 0) −1 3 −1 −1 (0, 1, 1) −1 −1 3 −1 (1, 0, 0) −1 3 −1 −1 (1, 0, 1) −1 −1 3 −1 (1, 1, 0) −1 −1 3 −1 (1, 1, 1) −1 −1 −1 3            

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 10 / 29

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Cube example part 2.

            1 1 1 1 2 8 24            

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 11 / 29

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Smith normal form.

Given any integer matrix A, we can perform row and column

  • perations so that:

A =        d1 · · · d2 · · · d3 · · · . . . . . . . . . ... . . . · · · dn        where d1|d2 , d2|d3, · · · , dn−1|dn.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 12 / 29

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Smith normal form.

Given any integer matrix A, we can perform row and column

  • perations so that:

A =        d1 · · · d2 · · · d3 · · · . . . . . . . . . ... . . . · · · dn        where d1|d2 , d2|d3, · · · , dn−1|dn. Invariant factors.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 12 / 29

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Smith normal form.

Given any integer matrix A, we can perform row and column

  • perations so that:

A =        d1 · · · d2 · · · d3 · · · . . . . . . . . . ... . . . · · · dn        where d1|d2 , d2|d3, · · · , dn−1|dn. Invariant factors. Elementary divisors.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 12 / 29

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Invariant factors vs. Elementary divisors

     1 3 6 60     

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 13 / 29

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Invariant factors vs. Elementary divisors

     1 3 6 60      Invariant factors: 1, 3, 6, 60

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 13 / 29

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Invariant factors vs. Elementary divisors

     1 3 6 60      Invariant factors: 1, 3, 6, 60      1 3 3 · 2 5 · 3 · 22     

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 13 / 29

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Invariant factors vs. Elementary divisors

     1 3 6 60      Invariant factors: 1, 3, 6, 60      1 3 3 · 2 5 · 3 · 22      Elementary divisors: 2, 22, 3, 3, 3, 5

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 13 / 29

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Back to the title.

Integer

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 14 / 29

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Back to the title.

Integer Invariants

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 14 / 29

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Back to the title.

Integer Invariants Abelian Cayley Graphs

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 14 / 29

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Back to the title.

Integer Invariants Abelian Cayley Graphs Graph: edges and vertices

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 14 / 29

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Back to the title.

Integer Invariants Abelian Cayley Graphs Graph: edges and vertices If vertices come from finite abelian group ⇒ abelian Cayley graph

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 14 / 29

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Group.

Group: set of elements with an operation (0, 0, 0) (1, 1, 1) (1, 0, 0) (1, 1, 0) (0, 1, 0) (0, 1, 1) (0, 0, 1) (1, 0, 1)

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 15 / 29

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Group.

Group: set of elements with an operation (0, 0, 0) (1, 1, 1) (1, 0, 0) (1, 1, 0) (0, 1, 0) (0, 1, 1) (0, 0, 1) (1, 0, 1) Closure Associativity Identity Inverses

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 15 / 29

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Connecting set E.

When is there an edge between two vertices?

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 16 / 29

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Connecting set E.

When is there an edge between two vertices? Define E.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 16 / 29

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Connecting set E.

When is there an edge between two vertices? Define E. E = {(0, 0, 1), (0, 1, 0), (1, 0, 0)}

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 16 / 29

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Connecting set E.

When is there an edge between two vertices? Define E. E = {(0, 0, 1), (0, 1, 0), (1, 0, 0)} Edge between g and h if g − h is in E.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 16 / 29

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Goals.

Predict the elementary divisors and their multiplicities for various incidence matrices.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 17 / 29

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Goals.

Predict the elementary divisors and their multiplicities for various incidence matrices. Recover the results of others in the field, but using a different technique.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 17 / 29

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Relationship between eigenvalues and Smith normal form.

Spectrum = eigenvalues and their multiplicities

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 18 / 29

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Relationship between eigenvalues and Smith normal form.

Spectrum = eigenvalues and their multiplicities A number λ is an eigenvalue of A if there exists a nonzero vector v such that Av = λv.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 18 / 29

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Relationship between eigenvalues and Smith normal form.

   2 2 2       2 1 2 1 2   

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 19 / 29

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Relationship between eigenvalues and Smith normal form.

   2 2 2    Eigenvalues: 2, 2, 2    2 1 2 1 2    Eigenvalues: 2, 2, 2

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 19 / 29

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SLIDE 39

Relationship between eigenvalues and Smith normal form.

   2 2 2    Eigenvalues: 2, 2, 2 SNF: 2, 2, 2    2 1 2 1 2    Eigenvalues: 2, 2, 2

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 19 / 29

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SLIDE 40

Relationship between eigenvalues and Smith normal form.

   2 2 2    Eigenvalues: 2, 2, 2 SNF: 2, 2, 2    2 1 2 1 2    Eigenvalues: 2, 2, 2 SNF: 1, 1, 8

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 19 / 29

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Relationship between eigenvalues and Smith normal form.

Theorem: For primes not dividing |G|, the multiplicity of pi as an elementary divisor of A, is the same as the number of eigenvalues exactly divisible by pi. [Sin, 2012]

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 20 / 29

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Eigenvalues as character Sums.

Theorem:

1 |G|MAM t = diag( e∈E

χ(e)) where χ ∈ Irred(G) [MacWilliams-Mann, 1968]

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 21 / 29

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Hamming association scheme H(n, q).

Let G be the set of all tuples of length n, with coordinates taken from some alphabet of size q.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 22 / 29

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Hamming association scheme H(n, q).

Let G be the set of all tuples of length n, with coordinates taken from some alphabet of size q. Two tuples are k-th associates if they differ in exactly k coordinate positions.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 22 / 29

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Hamming association scheme H(n, q).

Let G be the set of all tuples of length n, with coordinates taken from some alphabet of size q. Two tuples are k-th associates if they differ in exactly k coordinate positions. G = Zq × Zq × · · · × Zq (n times) Connecting set Ek: set of tuples with exactly k components that are not the identity.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 22 / 29

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Hamming association scheme H(n, q).

Let G be the set of all tuples of length n, with coordinates taken from some alphabet of size q. Two tuples are k-th associates if they differ in exactly k coordinate positions. G = Zq × Zq × · · · × Zq (n times) Connecting set Ek: set of tuples with exactly k components that are not the identity. The Hamming distance k between APPLE and SMILE is 3.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 22 / 29

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Hamming association scheme H(n, q).

Let G be the set of all tuples of length n, with coordinates taken from some alphabet of size q. Two tuples are k-th associates if they differ in exactly k coordinate positions. G = Zq × Zq × · · · × Zq (n times) Connecting set Ek: set of tuples with exactly k components that are not the identity. The Hamming distance k between APPLE and SMILE is 3. The Hamming distance between between MATH and COOL is 4.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 22 / 29

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Hamming association scheme H(n, q).

Let G be the set of all tuples of length n, with coordinates taken from some alphabet of size q. Two tuples are k-th associates if they differ in exactly k coordinate positions. G = Zq × Zq × · · · × Zq (n times) Connecting set Ek: set of tuples with exactly k components that are not the identity. The Hamming distance k between APPLE and SMILE is 3. The Hamming distance between between MATH and COOL is 4. The distance between (0, 1, 2, 2, 1) and (1, 1, 2, 2, 0) is 2.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 22 / 29

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SLIDE 49

Hamming association scheme H(n, q).

Let G be the set of all tuples of length n, with coordinates taken from some alphabet of size q. Two tuples are k-th associates if they differ in exactly k coordinate positions. G = Zq × Zq × · · · × Zq (n times) Connecting set Ek: set of tuples with exactly k components that are not the identity. The Hamming distance k between APPLE and SMILE is 3. The Hamming distance between between MATH and COOL is 4. The distance between (0, 1, 2, 2, 1) and (1, 1, 2, 2, 0) is 2. Construct adjacency matrix Ak.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 22 / 29

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Hamming association scheme H(n, q)

Can find the p-elementary divisor multiplicities for primes p not dividing |G|.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 23 / 29

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The n-cube graph.

Fix q = 2 and k = 1.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 24 / 29

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The n-cube graph.

Fix q = 2 and k = 1. Length n is the dimension you’re working in.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 24 / 29

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The n-cube graph.

Fix q = 2 and k = 1. Length n is the dimension you’re working in.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 24 / 29

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The n-cube graph.

Fix q = 2 and k = 1. Length n is the dimension you’re working in. G = Z2 × Z2 × · · · × Z2 (n times) Connecting set E1

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 24 / 29

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The n-cube graph.

Fix q = 2 and k = 1. Length n is the dimension you’re working in. G = Z2 × Z2 × · · · × Z2 (n times) Connecting set E1 When n is odd.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 24 / 29

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SLIDE 56

The n-cube graph.

Fix q = 2 and k = 1. Length n is the dimension you’re working in. G = Z2 × Z2 × · · · × Z2 (n times) Connecting set E1 When n is odd. When n is even.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 24 / 29

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SLIDE 57

When n is odd.

Let n = 3, the order of our group is 8.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 25 / 29

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SLIDE 58

When n is odd.

Let n = 3, the order of our group is 8. Eigenvalues: −n + 2j for 0 ≤ j ≤ n, with multiplicity n

j

  • .

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 25 / 29

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SLIDE 59

When n is odd.

Let n = 3, the order of our group is 8. Eigenvalues: −n + 2j for 0 ≤ j ≤ n, with multiplicity n

j

  • .

(Eigenvalue)(multiplicity) : −31, −13, 13, 31

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 25 / 29

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SLIDE 60

When n is odd.

Let n = 3, the order of our group is 8. Eigenvalues: −n + 2j for 0 ≤ j ≤ n, with multiplicity n

j

  • .

(Eigenvalue)(multiplicity) : −31, −13, 13, 31             1 1 1 1 1 1 3 3            

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 25 / 29

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SLIDE 61

When n is even.

Let n = 6, the order of our group is 64.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 26 / 29

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SLIDE 62

When n is even.

Let n = 6, the order of our group is 64. Eigenvalues: −n + 2j for 0 ≤ j ≤ n, with multiplicity n

j

  • .

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 26 / 29

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SLIDE 63

When n is even.

Let n = 6, the order of our group is 64. Eigenvalues: −n + 2j for 0 ≤ j ≤ n, with multiplicity n

j

  • .

(Eigenvalue)(multiplicity) : −61, −46, −215, 020, 215, 46, 61

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 26 / 29

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SLIDE 64

When n is even.

Let n = 6, the order of our group is 64. Eigenvalues: −n + 2j for 0 ≤ j ≤ n, with multiplicity n

j

  • .

(Eigenvalue)(multiplicity) : −61, −46, −215, 020, 215, 46, 61

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 26 / 29

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SLIDE 65

When n is even.

Let n = 6, the order of our group is 64. Eigenvalues: −n + 2j for 0 ≤ j ≤ n, with multiplicity n

j

  • .

(Eigenvalue)(multiplicity) : −61, −46, −215, 020, 215, 46, 61 Elementary divisors are 212, 32

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 26 / 29

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SLIDE 66

When n is even.

Let n = 6, the order of our group is 64. Eigenvalues: −n + 2j for 0 ≤ j ≤ n, with multiplicity n

j

  • .

(Eigenvalue)(multiplicity) : −61, −46, −215, 020, 215, 46, 61 Elementary divisors are 212, 32 Conjecture: The multiplicity of 2i as an elementary divisor is equal to the number of eigenvalues exactly divisible by 2i+1.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 26 / 29

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SLIDE 67

Generalizations.

G = Zq1 × Zq2 × · · · × Zqn Any connecting set E

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 27 / 29

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SLIDE 68

Generalizations.

G = Zq1 × Zq2 × · · · × Zqn Any connecting set E Laplacian Signless Laplacian Seidel

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 27 / 29

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SLIDE 69

Generalizations.

G = Zq1 × Zq2 × · · · × Zqn Any connecting set E Laplacian Signless Laplacian Seidel

  • H. Bai, 2002

Jacobson-Niedermaier-Reiner, 2003

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 27 / 29

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SLIDE 70

Acknowledgements.

Mentor Dr. Joshua Ducey

  • Dr. Minah Oh

Justin and Brock JMU Department of Mathematics and Statistics

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 28 / 29

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SLIDE 71

Questions?

125 × 125 incidence matrix for the parameters n = 3, q = 5, and k = 2.

Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 29 / 29