Counting Arithmetical Structures

Luis David Garćıa Puente

Department of Mathematics and StatisticsSam Houston State University

Blackwell-Tapia Conference 2018The Institute for Computational and Experimental Research in

MathematicsProvidence, RI

Statistical physics, a matrix and an abelian group

Gutenberg-Richter Law (1956)

The relationship between the magnitude M and total number N ofearthquakes in any given region and time period of magnitude≥ M is

N ∝ 10−bM , where b is a constant ∼ 1.

Statistical physics, a matrix and an abelian group

Gutenberg-Richter Law (1956)

The relationship between the magnitude M and total number N ofearthquakes in any given region and time period of magnitude≥ M is

N ∝ 10−bM , where b is a constant ∼ 1.

For each earthquake withmagnitude M ≥ 4 there areabout

I 0.1 with M ≥ 5I 0.01 with M ≥ 6I . . .

Gutenberg-Richter Law (in terms of Energy)

Shallow worldwide earthquakes 1976 - 2005 (Global CentroidMoment Tensor Project)

Gutenberg-Richter Law (in terms of Energy)

Shallow worldwide earthquakes 1977-2010 (Global CentroidMoment Tensor Project)

1. The Size of Earthquakes 11

• Shallow worldwide earthquakes (centroid moment tensor catalog):Conclusions 50

D(E)/ 1/E� ) ln D(E) = constant � � ln E

after Kagan

Geophys J Int 2002

Tectonophys 2010

Gutenberg-Richter Law

Per Bak (1996)

“This law is amazing! How canthe dynamics of all the elementsof a system as complicated as thecrust of the earth, withmountains, valleys, lakes, andgeological structures of enormousdiversity, conspire, as by magic,to produce a law with suchextreme simplicity?”

Self-organization towards criticality

Self-organized criticality (SOC) is a property of dynamicalsystems that have a critical point as an attractor. [Bak, Tang,Wiesenfeld (1987)]

Their macroscopic behavior displays the scale-invariancecharacteristic of the critical point of a phase transition, butwithout the need to tune control parameters to precise values.

This property is considered to be one of the mechanisms by whichcomplexity arises in nature.

It has been extensively studied in the statistical physics literatureduring the last three decades.

Mathematical Model for Sandpiles

In 1987, Bak, Tang, and Wiesenfeld proposed the following modelthat captures important features of self-organized criticality.

Model Structure and Definitions 3

Figure 1.2: The Bak, Tang and Wiesenfeld model of self-orgainized critical-ity.

observed in the way mean-squared velocity difference scales with distance,or in the spatial fractal structure of regions of high dissipation.

Systems that exhibit this significant correlation with power-law decayare said to have critical correlations. For equilibrium systems, such as ablock of ice, these critical correlations to power-law decay are replaced bya critical phase transition. Such critical points can only be achieved by ad-justing some physical parameter such as the temperature of the system.The behavior of a nonequilibrium system is independent of control param-eters and will achieve its critical state independent of any control parame-ter. Such a system is said to exhibit self-organized criticality. A system thatexhibits self-organized criticality will have a constant “input’ (steady ad-dition of sand grains), and a series of events or “avalanches” as “output”(sand avalanches). The “output” follows a power-law (fractal) frequency-size distribution. Self-organized criticality has been used to describe sys-tems such as forest fires, earthquakes, and stock-market fluctuations (Bak,1996). Understanding such systems, then, should lead to numerous physi-cal applications.

1.2 Model Structure and Definitions

The sandpile model was first generalized and developed by Dhar (1990).To understand his generalization, we recall some basic graph theory. Adirected graph Γ is a finite, nonempty set V (the vertex set) together with adisjoint set E (the edge set), possibly empty, that contains ordered pairs ofdistinct elements of V. An element of V is called a vertex and an element ofE is called an edge. Altogether, we denote the graph Γ(V, E). Let v1, v2 ∈ V.If there exists an edge (v1, v2) ∈ E, we say the vertices v1 and v2 are adjacent.

I The model is defined on a rectangulargrid of cells.

I The system evolves in discrete time.

I At each time step a sand grain isdropped onto a random grid cell.

I When a cell amasses four grains ofsand, it becomes unstable.

I It relaxes by toppling whereby foursand grains leave the site, and each ofthe four neighboring sites gets onegrain.

I This process continues until all sitesare stable.

Mathematical Model for Sandpiles

In 1987, Bak, Tang, and Wiesenfeld proposed the following modelthat captures important features of self-organized criticality.

Model Structure and Definitions 3

Figure 1.2: The Bak, Tang and Wiesenfeld model of self-orgainized critical-ity.

observed in the way mean-squared velocity difference scales with distance,or in the spatial fractal structure of regions of high dissipation.

Systems that exhibit this significant correlation with power-law decayare said to have critical correlations. For equilibrium systems, such as ablock of ice, these critical correlations to power-law decay are replaced bya critical phase transition. Such critical points can only be achieved by ad-justing some physical parameter such as the temperature of the system.The behavior of a nonequilibrium system is independent of control param-eters and will achieve its critical state independent of any control parame-ter. Such a system is said to exhibit self-organized criticality. A system thatexhibits self-organized criticality will have a constant “input’ (steady ad-dition of sand grains), and a series of events or “avalanches” as “output”(sand avalanches). The “output” follows a power-law (fractal) frequency-size distribution. Self-organized criticality has been used to describe sys-tems such as forest fires, earthquakes, and stock-market fluctuations (Bak,1996). Understanding such systems, then, should lead to numerous physi-cal applications.

1.2 Model Structure and Definitions

The sandpile model was first generalized and developed by Dhar (1990).To understand his generalization, we recall some basic graph theory. Adirected graph Γ is a finite, nonempty set V (the vertex set) together with adisjoint set E (the edge set), possibly empty, that contains ordered pairs ofdistinct elements of V. An element of V is called a vertex and an element ofE is called an edge. Altogether, we denote the graph Γ(V, E). Let v1, v2 ∈ V.If there exists an edge (v1, v2) ∈ E, we say the vertices v1 and v2 are adjacent.

I Start this process on an empty grid.

I At first there is little activity.

I As time goes on, the size (the totalnumber of topplings performed) of theavalanche caused by a single grain ofsand becomes hard to predict.

I A recurrent (or critical)configuration is a stable configurationthat appears infinitely often in thisMarkov process.

Distribution of sandpiles

Distribution of avalanche sizes in a 10×10 grid.

Abelian Sandpile Model on Graphs [Dhar (1990)]

Let Γ = (V ,E , s) denote a finite, con-nected, loopless multigraph with a dis-tinguished vertex s called the sink.

v1

s

v2

v3v4

A configuration over Γ is a function

σ : V \ {s} −→ N.

2

s

4

10

Stable Configurations

Given a graph Γ = (V ,E , s), for each v ∈ V \ {s} let dv be thenumber of edges incident to the vertex v .

DefinitionA configuration c is stable if and only if c(v) < dv for allv ∈ V \ {s}.

Stable Configurations

Given a graph Γ = (V ,E , s), for each v ∈ V \ {s} let dv be thenumber of edges incident to the vertex v .

DefinitionA configuration c is stable if and only if c(v) < dv for allv ∈ V \ {s}.

Stable Configurations

Given a graph Γ = (V ,E , s), for each v ∈ V \ {s} let dv be thenumber of edges incident to the vertex v .

DefinitionA configuration c is stable if and only if c(v) < dv for allv ∈ V \ {s}.

Toppling3

s

4

30

4

s

0

40

0

s

2

02

Stable Configurations

Given a graph Γ = (V ,E , s), for each v ∈ V \ {s} let dv be thenumber of edges incident to the vertex v .

DefinitionA configuration c is stable if and only if c(v) < dv for allv ∈ V \ {s}.

Toppling3

s

4

30

4

s

0

40

0

s

2

02

Stable Configurations

Given a graph Γ = (V ,E , s), for each v ∈ V \ {s} let dv be thenumber of edges incident to the vertex v .

DefinitionA configuration c is stable if and only if c(v) < dv for allv ∈ V \ {s}.

Toppling3

s

4

30

4

s

0

40

0

s

2

02

Sandpile Groups

Theorem (Dhar (1990))

Given a graph Γ = (V ,E , s). The set S(Γ) of recurrent sandpilestogether with stable addition forms a finite Abelian group, calledthe sandpile group of Γ.

Laplacian of a Graph

Given a graph Γ = (V ,E , s) with n + 1 vertices. The reducedLaplacian L̃ of Γ is the n×n matrix defined by

L̃ij =

di if i = j

−mij if ij ∈ E , with multiplicity mij0 otherwise

where we assume that the sink s is the (n + 1)-st vertex.

v1

s

v2

v3v4

L̃ =

4 −1 0 −1−1 4 −1 0

0 −1 4 −1−1 0 −1 4

.

Laplacian of a Graph

Given a graph Γ = (V ,E , s) with n + 1 vertices. The reducedLaplacian L̃ of Γ is the n×n matrix defined by

L̃ij =

di if i = j

−mij if ij ∈ E , with multiplicity mij0 otherwise

where we assume that the sink s is the (n + 1)-st vertex.

v1

s

v2

v3v4

L̃ =

4 −1 0 −1−1 4 −1 0

0 −1 4 −1−1 0 −1 4

.

Invariant Factors of the Sandpile Group S(Γ)

Theorem (Dhar (1990))

Given a graph Γ = (V ,E , s) with reduced Laplacian L̃. Letdiag(k1, k2, . . . , kn) be the Smith Normal Form of L̃. Then thesandpile group S(Γ) is isomorphic to

S(Γ) ∼= Zk1 × Zk2 × · · · × Zkn .

Invariant Factors of the Sandpile Group S(Γ)

Theorem (Dhar (1990))

Given a graph Γ = (V ,E , s) with reduced Laplacian L̃. Letdiag(k1, k2, . . . , kn) be the Smith Normal Form of L̃. Then thesandpile group S(Γ) is isomorphic to

S(Γ) ∼= Zk1 × Zk2 × · · · × Zkn .

v1

s

v2

v3v4

L̃ =

4 −1 0 −1−1 4 −1 0

0 −1 4 −1−1 0 −1 4

.

Invariant Factors of the Sandpile Group S(Γ)

Theorem (Dhar (1990))

Given a graph Γ = (V ,E , s) with reduced Laplacian L̃. Letdiag(k1, k2, . . . , kn) be the Smith Normal Form of L̃. Then thesandpile group S(Γ) is isomorphic to

S(Γ) ∼= Zk1 × Zk2 × · · · × Zkn .

v1

s

v2

v3v4

L̃ =

1 0 0 00 1 0 00 0 8 00 0 0 24

.

Invariant Factors of the Sandpile Group S(Γ)

Theorem (Dhar (1990))

Given a graph Γ = (V ,E , s) with reduced Laplacian L̃. Letdiag(k1, k2, . . . , kn) be the Smith Normal Form of L̃. Then thesandpile group S(Γ) is isomorphic to

S(Γ) ∼= Zk1 × Zk2 × · · · × Zkn .

v1

s

v2

v3v4

S(Γ) ∼= Z8 × Z24,|S(Γ)| = 8 · 24 = 192.

Invariant Factors of the Sandpile Group S(Γ)

Theorem (Dhar (1990))

Given a graph Γ = (V ,E , s) with reduced Laplacian L̃. Letdiag(k1, k2, . . . , kn) be the Smith Normal Form of L̃. Then thesandpile group S(Γ) is isomorphic to

S(Γ) ∼= Zk1 × Zk2 × · · · × Zkn .

Theorem (Matrix-Tree Theorem)

If Γ is a connected graph, then the number of spanning trees ofΓ, denoted κ(Γ), is equal to the determinant of the reducedLaplacian matrix L̃ of Γ. So

|S(Γ)| = κ(Γ).

Sandpiles on a grid Model Structure and Definitions 3

Figure 1.2: The Bak, Tang and Wiesenfeld model of self-orgainized critical-ity.

observed in the way mean-squared velocity difference scales with distance,or in the spatial fractal structure of regions of high dissipation.

Systems that exhibit this significant correlation with power-law decayare said to have critical correlations. For equilibrium systems, such as ablock of ice, these critical correlations to power-law decay are replaced bya critical phase transition. Such critical points can only be achieved by ad-justing some physical parameter such as the temperature of the system.The behavior of a nonequilibrium system is independent of control param-eters and will achieve its critical state independent of any control parame-ter. Such a system is said to exhibit self-organized criticality. A system thatexhibits self-organized criticality will have a constant “input’ (steady ad-dition of sand grains), and a series of events or “avalanches” as “output”(sand avalanches). The “output” follows a power-law (fractal) frequency-size distribution. Self-organized criticality has been used to describe sys-tems such as forest fires, earthquakes, and stock-market fluctuations (Bak,1996). Understanding such systems, then, should lead to numerous physi-cal applications.

1.2 Model Structure and Definitions

The sandpile model was first generalized and developed by Dhar (1990).To understand his generalization, we recall some basic graph theory. Adirected graph Γ is a finite, nonempty set V (the vertex set) together with adisjoint set E (the edge set), possibly empty, that contains ordered pairs ofdistinct elements of V. An element of V is called a vertex and an element ofE is called an edge. Altogether, we denote the graph Γ(V, E). Let v1, v2 ∈ V.If there exists an edge (v1, v2) ∈ E, we say the vertices v1 and v2 are adjacent.

I The sandpile group of the 2×2-grid is Z8 × Z24.

I The sandpile group of the 3×3-grid is Z4 × Z112 × Z224.

I The sandpile group of the 4×4-grid is Z28 × Z1320 × Z6600.

Sandpiles on a grid Model Structure and Definitions 3

Figure 1.2: The Bak, Tang and Wiesenfeld model of self-orgainized critical-ity.

observed in the way mean-squared velocity difference scales with distance,or in the spatial fractal structure of regions of high dissipation.

Systems that exhibit this significant correlation with power-law decayare said to have critical correlations. For equilibrium systems, such as ablock of ice, these critical correlations to power-law decay are replaced bya critical phase transition. Such critical points can only be achieved by ad-justing some physical parameter such as the temperature of the system.The behavior of a nonequilibrium system is independent of control param-eters and will achieve its critical state independent of any control parame-ter. Such a system is said to exhibit self-organized criticality. A system thatexhibits self-organized criticality will have a constant “input’ (steady ad-dition of sand grains), and a series of events or “avalanches” as “output”(sand avalanches). The “output” follows a power-law (fractal) frequency-size distribution. Self-organized criticality has been used to describe sys-tems such as forest fires, earthquakes, and stock-market fluctuations (Bak,1996). Understanding such systems, then, should lead to numerous physi-cal applications.

1.2 Model Structure and Definitions

The sandpile model was first generalized and developed by Dhar (1990).To understand his generalization, we recall some basic graph theory. Adirected graph Γ is a finite, nonempty set V (the vertex set) together with adisjoint set E (the edge set), possibly empty, that contains ordered pairs ofdistinct elements of V. An element of V is called a vertex and an element ofE is called an edge. Altogether, we denote the graph Γ(V, E). Let v1, v2 ∈ V.If there exists an edge (v1, v2) ∈ E, we say the vertices v1 and v2 are adjacent.

I The sandpile group of the 2×2-grid is Z8 × Z24.

I The sandpile group of the 3×3-grid is Z4 × Z112 × Z224.

I The sandpile group of the 4×4-grid is Z28 × Z1320 × Z6600.

Sandpiles on a grid

I The sandpile group of the 2×2-grid is Z8 × Z24.

I The sandpile group of the 3×3-grid is Z4 × Z112 × Z224.

I The sandpile group of the 4×4-grid is Z28 × Z1320 × Z6600.

Open Problem 1

Find a general formula for the sandpile group of an n×m-grid.

Open Problem 2

Give a complete characterization of the identity.

Identity sandpile in the 3276×3276 square grid

Color scheme: black=0, yellow=1, blue=2, and red=3.

Arithmetical Structures

I Let G be a finite, simple, connected graph with n ≥ 2 vertices.

I A be the adjacency matrix of G(Aij = # edges from vertex vi to vj .)

I An arithmetical structure of G is a pair (d, r) ∈ Zn>0 × Zn>0such that r is primitive (gcd of its coefficients = 1) and

(diag(d)− A) r = 0.

I Let D be the vertex-degree vector of G . Then diag(D)− A isthe Laplacian matrix of G and (D, 1̄) is an arithmeticalstructure.

Arithmetical Structures

I Let G be a finite, simple, connected graph with n ≥ 2 vertices.

I A be the adjacency matrix of G(Aij = # edges from vertex vi to vj .)

I An arithmetical structure of G is a pair (d, r) ∈ Zn>0 × Zn>0such that r is primitive (gcd of its coefficients = 1) and

(diag(d)− A) r = 0.

I Let D be the vertex-degree vector of G . Then diag(D)− A isthe Laplacian matrix of G and (D, 1̄) is an arithmeticalstructure.

Arithmetical Structures

I Let G be a finite, simple, connected graph with n ≥ 2 vertices.

I A be the adjacency matrix of G(Aij = # edges from vertex vi to vj .)

I An arithmetical structure of G is a pair (d, r) ∈ Zn>0 × Zn>0such that r is primitive (gcd of its coefficients = 1) and

(diag(d)− A) r = 0.

I Let D be the vertex-degree vector of G . Then diag(D)− A isthe Laplacian matrix of G and (D, 1̄) is an arithmeticalstructure.

Arithmetical Structures

I Let G be a finite, simple, connected graph with n ≥ 2 vertices.

I A be the adjacency matrix of G(Aij = # edges from vertex vi to vj .)

I An arithmetical structure of G is a pair (d, r) ∈ Zn>0 × Zn>0such that r is primitive (gcd of its coefficients = 1) and

(diag(d)− A) r = 0.

I Let D be the vertex-degree vector of G . Then diag(D)− A isthe Laplacian matrix of G and (D, 1̄) is an arithmeticalstructure.

Arithmetical Structures

Proposition (Lorenzini (1989))

The generalized Laplacian matrix L(G ,d) := diag(d)− A hasrank n − 1, and is an (almost non-singular) M-matrix:

Every (proper) principal minor of M is positive.

Corollary

d and r determines each other uniquely.

Arithmetical Structures

I Arith(G ) := {(d, r) | (d, r) is an arithmetical structure on G}.

I (G ,d, r) is called an arithmetical graph.

I coker L(G ,d) = Zn/ im L(G ,d) ∼= Z⊕K(G ,d, r).

I K(G ,d, r) is a finite abelian group called the critical group of(G ,d, r).

I K(G ,D, 1̄) is the sandpile group of G .

Arithmetical Structures

I Arith(G ) := {(d, r) | (d, r) is an arithmetical structure on G}.

I (G ,d, r) is called an arithmetical graph.

I coker L(G ,d) = Zn/ im L(G ,d) ∼= Z⊕K(G ,d, r).

I K(G ,d, r) is a finite abelian group called the critical group of(G ,d, r).

I K(G ,D, 1̄) is the sandpile group of G .

Arithmetical Structures

I Arith(G ) := {(d, r) | (d, r) is an arithmetical structure on G}.

I (G ,d, r) is called an arithmetical graph.

I coker L(G ,d) = Zn/ im L(G ,d) ∼= Z⊕K(G ,d, r).

I K(G ,d, r) is a finite abelian group called the critical group of(G ,d, r).

I K(G ,D, 1̄) is the sandpile group of G .

Arithmetical Structures

I Arith(G ) := {(d, r) | (d, r) is an arithmetical structure on G}.

I (G ,d, r) is called an arithmetical graph.

I coker L(G ,d) = Zn/ im L(G ,d) ∼= Z⊕K(G ,d, r).

I K(G ,d, r) is a finite abelian group called the critical group of(G ,d, r).

I K(G ,D, 1̄) is the sandpile group of G .

Arithmetical Structures

I Arith(G ) := {(d, r) | (d, r) is an arithmetical structure on G}.

I (G ,d, r) is called an arithmetical graph.

I coker L(G ,d) = Zn/ im L(G ,d) ∼= Z⊕K(G ,d, r).

I K(G ,d, r) is a finite abelian group called the critical group of(G ,d, r).

I K(G ,D, 1̄) is the sandpile group of G .

Arithmetical Structures

I Arith(G ) := {(d, r) | (d, r) is an arithmetical structure on G}.

I (G ,d, r) is called an arithmetical graph.

I coker L(G ,d) = Zn/ im L(G ,d) ∼= Z⊕K(G ,d, r).

I K(G ,d, r) is a finite abelian group called the critical group of(G ,d, r).

I K(G ,D, 1̄) is the sandpile group of G .

Theorem (Lorenzini ’89)

Arith(G ) is finite. ( Note: Proof is non-constructive.)

Motivation: Arithmetic Geometry (Lorenzini ’89)

I Let C be an algebraic curve that degenerates into ncomponents C1, . . . ,Cn.

I Let G = (V ,E ), where vertex vi corresponds to thecomponent Ci and |Ci ∩ Cj | = # edges from vi to vj .

I Let d be the vector of self-intersection numbers.

I The critical group K(G ,d, r) ∼= group of components of theNéron model of the Jacobian of C .

Motivation: Arithmetic Geometry (Lorenzini ’89)

I Let C be an algebraic curve that degenerates into ncomponents C1, . . . ,Cn.

I Let G = (V ,E ), where vertex vi corresponds to thecomponent Ci and |Ci ∩ Cj | = # edges from vi to vj .

I Let d be the vector of self-intersection numbers.

I The critical group K(G ,d, r) ∼= group of components of theNéron model of the Jacobian of C .

Motivation: Arithmetic Geometry (Lorenzini ’89)

I Let C be an algebraic curve that degenerates into ncomponents C1, . . . ,Cn.

I Let G = (V ,E ), where vertex vi corresponds to thecomponent Ci and |Ci ∩ Cj | = # edges from vi to vj .

I Let d be the vector of self-intersection numbers.

I The critical group K(G ,d, r) ∼= group of components of theNéron model of the Jacobian of C .

Motivation: Arithmetic Geometry (Lorenzini ’89)

I Let C be an algebraic curve that degenerates into ncomponents C1, . . . ,Cn.

I Let G = (V ,E ), where vertex vi corresponds to thecomponent Ci and |Ci ∩ Cj | = # edges from vi to vj .

I Let d be the vector of self-intersection numbers.

I The critical group K(G ,d, r) ∼= group of components of theNéron model of the Jacobian of C .

Arithmetical Structures on Paths

Proposition

In Pn, D is the only d-structure with di ≥ 2 for 1 < i < n.

Arithmetical Structures on Paths

Proposition

In Pn, D is the only d-structure with di ≥ 2 for 1 < i < n.

Theoremr = (r1, . . . , rn) ∈ Zn>0 is an r -structure on Pn if and only if

(i) r is primitive and r1 = rn = 1,

(ii) ri | (ri−1 + ri+1) for all i = 2, . . . , n − 1.

Arithmetical Structures on Paths

Proposition

In Pn, D is the only d-structure with di ≥ 2 for 1 < i < n.

Theoremr = (r1, . . . , rn) ∈ Zn>0 is an r -structure on Pn if and only if

(i) r is primitive and r1 = rn = 1,

(ii) ri | (ri−1 + ri+1) for all i = 2, . . . , n − 1.

D = (1, 1), r = (1, 1) (Laplacian a. s.).

Arithmetical Structures on Paths

Proposition

In Pn, D is the only d-structure with di ≥ 2 for 1 < i < n.

Theoremr = (r1, . . . , rn) ∈ Zn>0 is an r -structure on Pn if and only if

(i) r is primitive and r1 = rn = 1,

(ii) ri | (ri−1 + ri+1) for all i = 2, . . . , n − 1.

D = (1, 1), r = (1, 1) (Laplacian a. s.).

D = (1, 2, 1), r = (1, 1, 1)

d = (2, 1, 2), r = (1, 2, 1) 2 −1 0−1 1 −10 −1 2

121

= 0

Arithmetical Structures on Paths

Theoremr = (r1, . . . , rn) ∈ Zn>0 is an r -structure on Pn if and only if

(i) r is primitive and r1 = rn = 1,

(ii) ri | (ri−1 + ri+1) for all i = 2, . . . , n − 1.

D = (1, 1), r = (1, 1) (Laplacian a. s.).

D = (1, 2, 1), r = (1, 1, 1)

d = (2, 1, 2), r = (1, 2, 1)

D = (1, 2, 2, 1), r = (1, 1, 1, 1)

d = (2, 1, 3, 1), r = (1, 2, 1, 1)

d = (1, 3, 1, 2), r = (1, 1, 2, 1)

d = (3, 1, 2, 2), r = (1, 3, 2, 1)

d = (2, 2, 1, 3), r = (1, 2, 3, 1)

Arithmetical Structures on Cycles

Proposition

In Cn, D = 2̄ is the only d-structure with di ≥ 2 for all i .

Arithmetical Structures on Cycles

Proposition

In Cn, D = 2̄ is the only d-structure with di ≥ 2 for all i .

TheoremLet r = (r1, . . . , rn) ∈ Zn>0 be primitive. Then r is an r -structureon Cn if and only if

ri | (ri−1 + ri+1) for all i , with the indices taken modulo n.

Arithmetical Structures on Cycles

Proposition

In Cn, D = 2̄ is the only d-structure with di ≥ 2 for all i .

TheoremLet r = (r1, . . . , rn) ∈ Zn>0 be primitive. Then r is an r -structureon Cn if and only if

ri | (ri−1 + ri+1) for all i , with the indices taken modulo n.

D = (2, 2), r = (1, 1)

d = (1, 4), r = (2, 1)

d = (4, 1), r = (1, 2)

Arithmetical Structures on Cycles

TheoremLet r = (r1, . . . , rn) ∈ Zn>0 be primitive. Then r is an r -structureon Cn if and only if

ri | (ri−1 + ri+1) for all i , with the indices taken modulo n.

D = (2, 2), r = (1, 1)

d = (1, 4), r = (2, 1)

d = (4, 1), r = (1, 2)

D = (2, 2, 2), r = (1, 1, 1)

d = (3, 1, 3), r = (1, 2, 1) (3 permutations)

d = (2, 1, 5), r = (2, 3, 1) (6 permutations)

Counting Arithmetical Structures

Theorem (Ben Braun, Hugo Corrales, Scott Corry, ,Darren Glass, Nathan Kaplan, Jeremy Martin, Gregg MusikerCarlos Valencia, 2018)

Let Pn and Cn denote the path graph and cycle graph on nvertices, respectively. Then

|Arith(Pn)| = Cn−1 =1

n

(2n − 2n − 1

), |Arith(Cn)| = (2n−1)Cn−1 =

(2n − 1n − 1

).

K(Pn,d, r) = 0, K(Cn,d, r) = Zr(1), r(1) = #1’s in r .

Arithmetical Structures on Paths

Proposition

Given a triangulation T of an (n + 1)-gon, define

d(T ) = (d0, d1, . . . , dn) by di = # triangles incident to vertex i .

The map d gives a bijection between triangulations of an(n + 1)-gon and arithmetical d-structures on Pn.

Conway-Coxeter Frieze Patterns

Arithmetical Structures on Paths

Proposition

Given a triangulation T of an (n + 1)-gon, define

d(T ) = (d0, d1, . . . , dn) by di = # triangles incident to vertex i .

The map d gives a bijection between triangulations of an(n + 1)-gon and arithmetical d-structures on Pn.

Conway-Coxeter Frieze Patterns

Figure 2: John Horton Conway (left) (born 1937) and Harold Scott MacDonaldCoxeter (right) (1907 – 2003).

Harold Scott MacDonald Coxeter (1907 – 2003) (see Figure 2), who intro-duced and studied them in the early 1970s [5]. Before we are going to definefrieze patterns properly, you might want to have a look at some first exam-ples in Figure 3 (the green [italic] and red [bold] colours will be explained later).

1 1 1 1 1 1 1 1 1 1 1 1 1 1· · · 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 · · ·

1 1 1 1 1 1 1 1 1 1 1 1 1 1

(a) A simple frieze pattern.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1· · · 2 2 1 4 2 1 4 1 3 2 1 4 2 1 4 1 3 2 · · ·

5 1 3 7 1 3 3 2 5 1 3 7 1 3 3 2 5· · · 3 2 2 5 3 2 2 5 3 2 2 5 3 2 2 5 3 2 · · ·

1 3 3 2 5 1 3 7 1 3 3 2 5 1 3 7 1· · · 2 1 4 1 3 2 1 4 2 1 4 1 3 2 1 4 2 1 · · ·

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(b) A more complicated frieze pattern.

Figure 3: First examples of frieze patterns.

2

Arithmetical Structures on Paths

Proposition

Given a triangulation T of an (n + 1)-gon, define

d(T ) = (d0, d1, . . . , dn) by di = # triangles incident to vertex i .

The map d gives a bijection between triangulations of an(n + 1)-gon and arithmetical d-structures on Pn.

Conway-Coxeter Frieze Patterns

Arithmetical Structures on Paths

Proposition

Given a triangulation T of an (n + 1)-gon, define

d(T ) = (d0, d1, . . . , dn) by di = # triangles incident to vertex i .

The map d gives a bijection between triangulations of an(n + 1)-gon and arithmetical d-structures on Pn.

Conway-Coxeter Frieze Patterns

Figure 2: John Horton Conway (left) (born 1937) and Harold Scott MacDonaldCoxeter (right) (1907 – 2003).

Harold Scott MacDonald Coxeter (1907 – 2003) (see Figure 2), who intro-duced and studied them in the early 1970s [5]. Before we are going to definefrieze patterns properly, you might want to have a look at some first exam-ples in Figure 3 (the green [italic] and red [bold] colours will be explained later).

1 1 1 1 1 1 1 1 1 1 1 1 1 1· · · 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 · · ·

1 1 1 1 1 1 1 1 1 1 1 1 1 1

(a) A simple frieze pattern.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1· · · 2 2 1 4 2 1 4 1 3 2 1 4 2 1 4 1 3 2 · · ·

5 1 3 7 1 3 3 2 5 1 3 7 1 3 3 2 5· · · 3 2 2 5 3 2 2 5 3 2 2 5 3 2 2 5 3 2 · · ·

1 3 3 2 5 1 3 7 1 3 3 2 5 1 3 7 1· · · 2 1 4 1 3 2 1 4 2 1 4 1 3 2 1 4 2 1 · · ·

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(b) A more complicated frieze pattern.

Figure 3: First examples of frieze patterns.

2

Arithmetical Structures on Stars

Let Kn,1 denote the star graph with n leaves.

TheoremThe d-arithmetical structures on Kn,1 are the positive integersolutions to

d0 =n∑

i=1

1

di.

Each such solution is an Egyptian fraction representation of d0.

Observation: There is no closed form for the sequence|Arith(Kn,1)|.

1, 2, 14, 263, 13462, 2104021, . . .

Arithmetical Structures on Dynkin Graphs

Arithmetical structures on bidentsKassie Archer, Abigail C. Bishop, Alexander Diaz-Lopez,

Luis D. García Puente, Darren Glass, Joel Louwsma

x

y

v0 v1· · ·

v¸

rx

ry

r0 r1· · ·

r¸

1 IntroductionAn arithmetical structure on a graph is typically given by a pair of vectors r, d so that

We note that the existence of a unique vector r whose entries have greatest common divisorequal to one follows from the fact that the nullspace of A ≠ D is one dimensional. While thischoice of r is standard in the literature, we often find it more convenient to instead choosethe vector scaled by a factor of r0rxry . It is straightforward to check that we then have thatr̄0 = r0 r0rxry = rx

r0rxry

· ry r0rx ry = r̄xry. In other words, the structure takes the form:

a

b

ab r̄1· · ·

r̄¸

One advantage of looking at r̄-structures is that for every pair of positive integers a, b there willbe a unique choice of n and a unique arithmetical structure with r̄x = a and r̄y = b. We will seethis more explicitly in Theorem 3.3

2 Smooth arithmetical structuresIn this section, we will define the notion of smooth arithmetical structure and discuss how allarithmetical structures on Dn can be understood in terms of smooth arithmetical structures on Dnand certain smaller graphs.

2.1 Definition and basic propertiesWe say that an arithmetical structure (d, r) on Dn, n Ø 4, is smooth if dx, dy, d1, d2, . . . , d¸ Ø 2. Itwill be convenient to extend this notion to certain smaller graphs in the following way. We saythat an arithmetical structure on D3 is smooth if dx, dy Ø 2. Let Pk,1, k Ø 2, be the path with k

1

Theorem (Kassie Archer, Abigail Bishop, AlexanderDiaz-Lopez, , Darren Glass, Joel Lowsma (2018))

Let Dn denote the Dynkin graph with n vertices. Then

14Cn−3 ≤ |Arith(Dn)| ≤Cn−3

3(4n4 − 20n3 + 38n2 − 28n − 36),

where Cn is the nth Catalan number.Moreover, |K(Dn; d, r)| is a cyclic group of order

|K(Dn; d, r)| =r0

rx ry r`.