How to build a neighbor table for the Hexagonal lattice

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I want to consider a collection of vertices arranged in a finite Hexagonal lattice, with say $n_r$ rows and $n_x$ vertices per row, for a total of $N = n_xn_r$ vertices. The goal is to construct a neighbor table, which is a matrix $A$ with dimensionality $Ntimes N$. Each element $A_ij$ is one if there is a bond connecting vertices $i$ and $j$, and zero otherwise. This can also be seen as an adjacency matrix for a particular graph.

I can't think of a way to build this matrix that isn't frustrating to code as a function of $n_r$ and $n_x$, but I'm aware there are some lattice functions built into Mathematica. Maybe those can make the process more smooth if anyone has suggestions.

Also, to make things a little harder, I also want the option of including periodic boundary conditions. This just means if you walk off the finite lattice in a particular direction (say, off the left-hand side) you reappear on the opposite side of the lattice (in this case the right hand side). This means there are new connections that would look long-ranged in any planar representation of the graph, or you can think of it as putting the graph on a torus. Help with that case would be particularly appreciated.

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asked 3 hours ago

mflynn

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favorite

asked 3 hours ago

mflynn

1084

graphs-and-networks lattices adjacency

asked 3 hours ago

mflynn

1084

asked 3 hours ago

mflynn

1084

asked 3 hours ago

mflynn

1084

asked 3 hours ago

mflynn

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asked 3 hours ago

mflynn

1084

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1 Answer
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I will take this opportunity to showcase the abilities of IGraph/M for lattice generation and mesh / graph / matrix conversions. IGraph/M thrives on user feedback, so if you find it useful, please take some time to write a few comments about you experiece. It will help me improve the package.

Non-periodic case

You can directly generate a (non-periodic) lattice with IGraph/M.

<< IGraphM`

mesh = IGLatticeMesh["Hexagonal", Polygon@CirclePoints[3, 6], 
 MeshCellLabel -> 2 -> "Index"]

enter image description here

The second argument of IGLatticeMesh may be a region. This region will be filled with cells. In this case, I chose a big hexagon to be filled with small hexagonal cells.

The cell adjacency matrix:

am = IGMeshCellAdjacencyMatrix[mesh, 2]

"2" means 2-dimensional cells, i.e. little hexagons. "1" would mean edges and "0" points.

MatrixPlot[am]

enter image description here

If you need the graph,

graph = IGMeshCellAdjacencyGraph[mesh, 2, 
 VertexCoordinates -> Automatic]

enter image description here

Notice that this is actually a triangular connectivity, which could also be generated directly (in some shapes) with IGTriangularLattice. Demo:

IGTriangularLattice[4], IGTriangularLattice[3, 5]

enter image description here

We could have use IGLatticeMesh too:

IGLatticeMesh["Triangular", 3, 3]

enter image description here

Let's get the point-to-point connectivity now (instead of the cell-to-cell one):

IGMeshCellAdjacencyGraph[%, 0]

enter image description here

Periodic case

Now let us do the periodic case.

We start with a hex lattice arrange in an $ntimes m$ grid.

n, m = 5, 6;
mesh = IGLatticeMesh["Hexagonal", n, m, MeshCellLabel -> 2 -> "Index"]

enter image description here

Convert it to a graph. This time I will not preserve the vertex coordinates so that we can get a clearer layout after we make the lattice periodic.

graph = IGMeshCellAdjacencyGraph[mesh, 2, VertexLabels -> "Name"];
graph = VertexReplace[graph, 2, i_ :> i]

I have also converted the vertex names, which were of the form 2, index (2 indicating two-dimensional mesh cells) to simply index.

enter image description here

We add the extra edges needed for periodic boundary conditions.

extraEdges = DeleteDuplicates@Flatten@Table[
 If[Mod[i, m] == 0, i <-> i - m + 1, i <-> Mod[i - 2 m + 1, m n, 1], ],
 If[i <= m, i <-> i + m n - m, i <-> Mod[i + m n - m + 1, m n, 1], ],
 i, m n
 ]

pgraph = EdgeAdd[graph, extraEdges]

Then we can get (or plot) the graph's adjacency matrix.

IGAdjacencyMatrixPlot[pgraph]

enter image description here

am = AdjacencyMatrix[graph]

Extra visualization: here's the graph in 3D with m,n = 10,20:

(* remember to re-create graph and extraEdges after setting m,n *)
pgraph = Graph3D[EdgeAdd[graph, extraEdges], VertexLabels -> None]

enter image description here

An alternative solution for the periodic case

The adjacency relations of hexagonal cells form a triangular lattice. There is a function in IGraph/M for directly generating a triangular lattice graph, and it has an option to make it periodic:

IGTriangularLattice[5, 10]

enter image description here

IGTriangularLattice[5, 10, "Periodic" -> True]

enter image description here

Then you can just get the adjacency matrix again.

Note that the m,n syntax in IGLatticeMesh and IGTriangularLattice do not have the exact same meaningÃ¢Â€Â”pay attention to the difference if you mix these approaches! The vertex labelling will also be different. Presumably at some point you will want to use the visualization of the hex lattice mesh to plot your results. Thus it is useful to be able to map back to mesh cell indices.

edited 15 mins ago

answered 57 mins ago

Szabolcs

155k13419909

add a commentÂ |Â

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1 Answer
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active

oldest

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up vote
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accepted

Non-periodic case

You can directly generate a (non-periodic) lattice with IGraph/M.

<< IGraphM`

mesh = IGLatticeMesh["Hexagonal", Polygon@CirclePoints[3, 6], 
 MeshCellLabel -> 2 -> "Index"]

enter image description here

The second argument of IGLatticeMesh may be a region. This region will be filled with cells. In this case, I chose a big hexagon to be filled with small hexagonal cells.

The cell adjacency matrix:

am = IGMeshCellAdjacencyMatrix[mesh, 2]

"2" means 2-dimensional cells, i.e. little hexagons. "1" would mean edges and "0" points.

MatrixPlot[am]

enter image description here

If you need the graph,

graph = IGMeshCellAdjacencyGraph[mesh, 2, 
 VertexCoordinates -> Automatic]

enter image description here

Notice that this is actually a triangular connectivity, which could also be generated directly (in some shapes) with IGTriangularLattice. Demo:

IGTriangularLattice[4], IGTriangularLattice[3, 5]

enter image description here

We could have use IGLatticeMesh too:

IGLatticeMesh["Triangular", 3, 3]

enter image description here

Let's get the point-to-point connectivity now (instead of the cell-to-cell one):

IGMeshCellAdjacencyGraph[%, 0]

enter image description here

Periodic case

Now let us do the periodic case.

We start with a hex lattice arrange in an $ntimes m$ grid.

n, m = 5, 6;
mesh = IGLatticeMesh["Hexagonal", n, m, MeshCellLabel -> 2 -> "Index"]

enter image description here

Convert it to a graph. This time I will not preserve the vertex coordinates so that we can get a clearer layout after we make the lattice periodic.

graph = IGMeshCellAdjacencyGraph[mesh, 2, VertexLabels -> "Name"];
graph = VertexReplace[graph, 2, i_ :> i]

I have also converted the vertex names, which were of the form 2, index (2 indicating two-dimensional mesh cells) to simply index.

enter image description here

We add the extra edges needed for periodic boundary conditions.

extraEdges = DeleteDuplicates@Flatten@Table[
 If[Mod[i, m] == 0, i <-> i - m + 1, i <-> Mod[i - 2 m + 1, m n, 1], ],
 If[i <= m, i <-> i + m n - m, i <-> Mod[i + m n - m + 1, m n, 1], ],
 i, m n
 ]

pgraph = EdgeAdd[graph, extraEdges]

Then we can get (or plot) the graph's adjacency matrix.

IGAdjacencyMatrixPlot[pgraph]

enter image description here

am = AdjacencyMatrix[graph]

Extra visualization: here's the graph in 3D with m,n = 10,20:

(* remember to re-create graph and extraEdges after setting m,n *)
pgraph = Graph3D[EdgeAdd[graph, extraEdges], VertexLabels -> None]

enter image description here

An alternative solution for the periodic case

The adjacency relations of hexagonal cells form a triangular lattice. There is a function in IGraph/M for directly generating a triangular lattice graph, and it has an option to make it periodic:

IGTriangularLattice[5, 10]

enter image description here

IGTriangularLattice[5, 10, "Periodic" -> True]

enter image description here

Then you can just get the adjacency matrix again.

edited 15 mins ago

answered 57 mins ago

Szabolcs

155k13419909

add a commentÂ |Â

up vote
4
down vote

accepted

Non-periodic case

You can directly generate a (non-periodic) lattice with IGraph/M.

<< IGraphM`

mesh = IGLatticeMesh["Hexagonal", Polygon@CirclePoints[3, 6], 
 MeshCellLabel -> 2 -> "Index"]

enter image description here

The second argument of IGLatticeMesh may be a region. This region will be filled with cells. In this case, I chose a big hexagon to be filled with small hexagonal cells.

The cell adjacency matrix:

am = IGMeshCellAdjacencyMatrix[mesh, 2]

"2" means 2-dimensional cells, i.e. little hexagons. "1" would mean edges and "0" points.

MatrixPlot[am]

enter image description here

If you need the graph,

graph = IGMeshCellAdjacencyGraph[mesh, 2, 
 VertexCoordinates -> Automatic]

enter image description here

Notice that this is actually a triangular connectivity, which could also be generated directly (in some shapes) with IGTriangularLattice. Demo:

IGTriangularLattice[4], IGTriangularLattice[3, 5]

enter image description here

We could have use IGLatticeMesh too:

IGLatticeMesh["Triangular", 3, 3]

enter image description here

Let's get the point-to-point connectivity now (instead of the cell-to-cell one):

IGMeshCellAdjacencyGraph[%, 0]

enter image description here

Periodic case

Now let us do the periodic case.

We start with a hex lattice arrange in an $ntimes m$ grid.

n, m = 5, 6;
mesh = IGLatticeMesh["Hexagonal", n, m, MeshCellLabel -> 2 -> "Index"]

enter image description here

Convert it to a graph. This time I will not preserve the vertex coordinates so that we can get a clearer layout after we make the lattice periodic.

graph = IGMeshCellAdjacencyGraph[mesh, 2, VertexLabels -> "Name"];
graph = VertexReplace[graph, 2, i_ :> i]

I have also converted the vertex names, which were of the form 2, index (2 indicating two-dimensional mesh cells) to simply index.

enter image description here

We add the extra edges needed for periodic boundary conditions.

extraEdges = DeleteDuplicates@Flatten@Table[
 If[Mod[i, m] == 0, i <-> i - m + 1, i <-> Mod[i - 2 m + 1, m n, 1], ],
 If[i <= m, i <-> i + m n - m, i <-> Mod[i + m n - m + 1, m n, 1], ],
 i, m n
 ]

pgraph = EdgeAdd[graph, extraEdges]

Then we can get (or plot) the graph's adjacency matrix.

IGAdjacencyMatrixPlot[pgraph]

enter image description here

am = AdjacencyMatrix[graph]

Extra visualization: here's the graph in 3D with m,n = 10,20:

(* remember to re-create graph and extraEdges after setting m,n *)
pgraph = Graph3D[EdgeAdd[graph, extraEdges], VertexLabels -> None]

enter image description here

An alternative solution for the periodic case

The adjacency relations of hexagonal cells form a triangular lattice. There is a function in IGraph/M for directly generating a triangular lattice graph, and it has an option to make it periodic:

IGTriangularLattice[5, 10]

enter image description here

IGTriangularLattice[5, 10, "Periodic" -> True]

enter image description here

Then you can just get the adjacency matrix again.

edited 15 mins ago

answered 57 mins ago

Szabolcs

155k13419909

add a commentÂ |Â

up vote
4
down vote

accepted

Non-periodic case

You can directly generate a (non-periodic) lattice with IGraph/M.

<< IGraphM`

mesh = IGLatticeMesh["Hexagonal", Polygon@CirclePoints[3, 6], 
 MeshCellLabel -> 2 -> "Index"]

enter image description here

The second argument of IGLatticeMesh may be a region. This region will be filled with cells. In this case, I chose a big hexagon to be filled with small hexagonal cells.

The cell adjacency matrix:

am = IGMeshCellAdjacencyMatrix[mesh, 2]

"2" means 2-dimensional cells, i.e. little hexagons. "1" would mean edges and "0" points.

MatrixPlot[am]

enter image description here

If you need the graph,

graph = IGMeshCellAdjacencyGraph[mesh, 2, 
 VertexCoordinates -> Automatic]

enter image description here

Notice that this is actually a triangular connectivity, which could also be generated directly (in some shapes) with IGTriangularLattice. Demo:

IGTriangularLattice[4], IGTriangularLattice[3, 5]

enter image description here

We could have use IGLatticeMesh too:

IGLatticeMesh["Triangular", 3, 3]

enter image description here

Let's get the point-to-point connectivity now (instead of the cell-to-cell one):

IGMeshCellAdjacencyGraph[%, 0]

enter image description here

Periodic case

Now let us do the periodic case.

We start with a hex lattice arrange in an $ntimes m$ grid.

n, m = 5, 6;
mesh = IGLatticeMesh["Hexagonal", n, m, MeshCellLabel -> 2 -> "Index"]

enter image description here

Convert it to a graph. This time I will not preserve the vertex coordinates so that we can get a clearer layout after we make the lattice periodic.

graph = IGMeshCellAdjacencyGraph[mesh, 2, VertexLabels -> "Name"];
graph = VertexReplace[graph, 2, i_ :> i]

I have also converted the vertex names, which were of the form 2, index (2 indicating two-dimensional mesh cells) to simply index.

enter image description here

We add the extra edges needed for periodic boundary conditions.

extraEdges = DeleteDuplicates@Flatten@Table[
 If[Mod[i, m] == 0, i <-> i - m + 1, i <-> Mod[i - 2 m + 1, m n, 1], ],
 If[i <= m, i <-> i + m n - m, i <-> Mod[i + m n - m + 1, m n, 1], ],
 i, m n
 ]

pgraph = EdgeAdd[graph, extraEdges]

Then we can get (or plot) the graph's adjacency matrix.

IGAdjacencyMatrixPlot[pgraph]

enter image description here

am = AdjacencyMatrix[graph]

Extra visualization: here's the graph in 3D with m,n = 10,20:

(* remember to re-create graph and extraEdges after setting m,n *)
pgraph = Graph3D[EdgeAdd[graph, extraEdges], VertexLabels -> None]

enter image description here

An alternative solution for the periodic case

The adjacency relations of hexagonal cells form a triangular lattice. There is a function in IGraph/M for directly generating a triangular lattice graph, and it has an option to make it periodic:

IGTriangularLattice[5, 10]

enter image description here

IGTriangularLattice[5, 10, "Periodic" -> True]

enter image description here

Then you can just get the adjacency matrix again.

edited 15 mins ago

answered 57 mins ago

Szabolcs

155k13419909

Non-periodic case

You can directly generate a (non-periodic) lattice with IGraph/M.

<< IGraphM`

mesh = IGLatticeMesh["Hexagonal", Polygon@CirclePoints[3, 6], 
 MeshCellLabel -> 2 -> "Index"]

enter image description here

The second argument of IGLatticeMesh may be a region. This region will be filled with cells. In this case, I chose a big hexagon to be filled with small hexagonal cells.

The cell adjacency matrix:

am = IGMeshCellAdjacencyMatrix[mesh, 2]

"2" means 2-dimensional cells, i.e. little hexagons. "1" would mean edges and "0" points.

MatrixPlot[am]

enter image description here

If you need the graph,

graph = IGMeshCellAdjacencyGraph[mesh, 2, 
 VertexCoordinates -> Automatic]

enter image description here

Notice that this is actually a triangular connectivity, which could also be generated directly (in some shapes) with IGTriangularLattice. Demo:

IGTriangularLattice[4], IGTriangularLattice[3, 5]

enter image description here

We could have use IGLatticeMesh too:

IGLatticeMesh["Triangular", 3, 3]

enter image description here

Let's get the point-to-point connectivity now (instead of the cell-to-cell one):

IGMeshCellAdjacencyGraph[%, 0]

enter image description here

Periodic case

Now let us do the periodic case.

We start with a hex lattice arrange in an $ntimes m$ grid.

n, m = 5, 6;
mesh = IGLatticeMesh["Hexagonal", n, m, MeshCellLabel -> 2 -> "Index"]

enter image description here

Convert it to a graph. This time I will not preserve the vertex coordinates so that we can get a clearer layout after we make the lattice periodic.

graph = IGMeshCellAdjacencyGraph[mesh, 2, VertexLabels -> "Name"];
graph = VertexReplace[graph, 2, i_ :> i]

I have also converted the vertex names, which were of the form 2, index (2 indicating two-dimensional mesh cells) to simply index.

enter image description here

We add the extra edges needed for periodic boundary conditions.

extraEdges = DeleteDuplicates@Flatten@Table[
 If[Mod[i, m] == 0, i <-> i - m + 1, i <-> Mod[i - 2 m + 1, m n, 1], ],
 If[i <= m, i <-> i + m n - m, i <-> Mod[i + m n - m + 1, m n, 1], ],
 i, m n
 ]

pgraph = EdgeAdd[graph, extraEdges]

Then we can get (or plot) the graph's adjacency matrix.

IGAdjacencyMatrixPlot[pgraph]

enter image description here

am = AdjacencyMatrix[graph]

Extra visualization: here's the graph in 3D with m,n = 10,20:

(* remember to re-create graph and extraEdges after setting m,n *)
pgraph = Graph3D[EdgeAdd[graph, extraEdges], VertexLabels -> None]

enter image description here

An alternative solution for the periodic case

The adjacency relations of hexagonal cells form a triangular lattice. There is a function in IGraph/M for directly generating a triangular lattice graph, and it has an option to make it periodic:

IGTriangularLattice[5, 10]

enter image description here

IGTriangularLattice[5, 10, "Periodic" -> True]

enter image description here

Then you can just get the adjacency matrix again.

edited 15 mins ago

answered 57 mins ago

Szabolcs

155k13419909

edited 15 mins ago

answered 57 mins ago

Szabolcs

155k13419909

answered 57 mins ago

Szabolcs

155k13419909

answered 57 mins ago

Szabolcs

155k13419909

add a commentÂ |Â

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