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Thanks to Mark Newman and Christian Pich for providing the following

The simplest representation uses a square grid, with additional
diagonal edges one way across each square.  Thus the grid  element
(i,j) is a neighbor of:

(i+1,j)
(i-1,j)
(i,j+1)
(i,j-1)
(i+1,j+1)
(i-1,j-1)

In this way, enumerating vertices in row-major order (i.e., all of the
first lattice row, then all of the second row, and so forth), the
adjacency matrix of a triangular grid of n^2 elements (without
periodical suturing) is a matrix of order n^2, containing n^2 blocks
each-one of order n. Among all these blocks, the nonzero blocks are
the following:

(i) n diagonal blocks with 1s only just above and below the diagonal.

(ii) n - 1 blocks just above the diagonal blocks, which are starting
from the block position (1,2) and going up to the block position
(n-1,n) and each one includes 1s only on its diagonal and just above
the diagonal.

(iii) n - 1 blocks just below the diagonal blocks, which are starting
from the block position (2,1) and going up to the block position
(n,n-1) and each one includes 1s only on its diagonal and just below
the diagonal.

Best,

--Moses

On Thu, Nov 5, 2009 at 6:19 PM, Moses Boudourides
> Hello,
>
> Say we have a triangular grid (tiling) formed by nxm vertices (i.e., a
> graph in which all internal vertices have degree 6). What I want to
> obtain is the structure of the matrix (or list) representation of its
> edges. Of course, this depends on the enumeration of vertices. Thus,
> the question is what re-ordering gives the simplest general form of
> such a matrix (or list) and what is exactly this form?
>
> My motivation comes from that I want to make a simulation of a network
> influence situation in which edges (ties) are influenced by the
> incident ones (than vertices being influenced by the neighboring ones)
> according to certain rules. In fact, first I'm assuming that the grid
> is signed and that signs of edges may change in such a way that they
> follow the structural stability or instability of the triangles with
> common edges.
>
> I would appreciate any help or hints on how I might proceed in dealing
> with this problem.
>
> Best,
>
> --Moses
>

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