Adjacency matrix for a square lattice

I am trying to create an adjacency matrix for $n\times n$ square lattice. If $(i,j)$ denotes a vertex in the lattice then I first index all the nodes in the lattice using a single index $k$ such that $k=(i-1)*n+j$ . After that I want to write $n^2\times n^2$ adjacency matrix. But I am not able to do it properly. Please help me. Thank you

Easy Math Editor

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`2 \times 3`	$2 \times 3$
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Comments

You move between adjacent vertices in the same row by adding or subtracting $1$ , and you move between adjacent vertices in the same column by adding or subtracting $n$ . Thus vertex $k$ is adjacent to:

Vertex $k+1$ , unless $n$ divides $k$ ,
Vertex $k-1$ , unless $n$ divides $k-1$ ,
Vertex $k+n$ , unless $k+n > n^2$ ,
Vertex $k-n$ , unless $k-n\le 0$ .

Adjacency matrices for $2\times2$ and $3\times3$ grids are $\left(\begin{array}{cccc}0&1&1&0\\1&0&0&1\\1&0&0&1\\0&1&1&0\end{array}\right) \qquad \qquad \left(\begin{array}{ccccccccc}0&1&0&1&0&0&0&0&0\\1&0&1&0&1&0&0&0&0\\0&1&0&0&0&1&0&0&0\\ 1&0&0&0&1&0&1&0&0\\0&1&0&1&0&1&0&1&0\\0&0&1&0&1&0&0&0&1\\ 0&0&0&1&0&0&0&1&0\\0&0&0&0&1&0&1&0&1\\0&0&0&0&0&1&0&1&0 \end{array}\right)$

Mark Hennings - 7 years, 8 months ago

Thank you. This really looks cool.. I will modify my code now. I was mainly having problem at edges but now I got your idea of checking whether $n$ divides those indices or not. I will get back to you soon.

Snehal Shekatkar - 7 years, 8 months ago

Yess.. I modified my code and it works perfectly now!! Thank you very much sir!

Snehal Shekatkar - 7 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$