Let us consider the matrix $S$ of a Sudoku, $S$ being $n\times n$,
with $n$ being a square (i.e. 9, 16, 25, etc.).
Let's say that $S_{i,j}:=S_{i,j}\pmod{2}$, i.e. we now consider $S$
with all its elements being taken in $1/\mathbb Z_2$ (i.e. the 2-adic
numbers), for $1\leq i\leq n$, $1\leq j\leq n$.

We now consider $S$ as the adjacency matrix of a graph $G$.

I know have the two following questions (we consider that $S$ is the
previous $S$ taken in $1/\mathbb Z_2$, as explained before):

1. The graph done from $S$ is often not simple, as it contains at
least one loop ($S$ has not often all its diagonal elements equalling
$0$). But what can we say about ``often?:'' are there correct (and
evidently completed) Sudoku matrices leading to, for their
representation in $1/\mathbb Z_2$, a diagonal full of zeros (and thus
a trace equalling $0$)?

2. Is it possible for $S$ to be symmetric, the previous $S$ being a
correct (and evidently completed) Sudoku matrix? If it is so, G is non-
oriented.

It is difficult to answer correctly to these questions without any
numerical (i.e. computing) help.

Is there somebody here having propositions about these two problems?

* This is not $1/\mathbb Z$, but $\mathbb Z/2\mathbb Z$,
* By "2-adic numbers," I was speaking (erroneously) about the integers
modulo 2. Evidently, the $S_{i,j}$ are taken modulo 2.

For odd n, using the labelling starting from 1 to n,
there will be an odd number of 1's in the resulting 0-1
matrix, so if there is a sudoku that has a symmetric
derived 0-1 matrix, that matrix will have a non-zero
trace.  This difficulty disappears if the labels range
from 0 to n-1.

One can use various constructions for matrices to build
larger matrices, such as Hadamard product.  Given two
p by p matrices (or non-square) A and B, A prod B has
something like a_ij*b_kl for the entry in the i*p+k row
and j*p +l column.  The matrix above can be used in a
similar fashion to create a larger matrix if you replace
a_ij*b_kl  with (a_ij,b_kl), a base p representation for
the number from 1 to p^2.  Details need to be worked out
to encode the desired result correctly.  With such a
construction, I believe that the answer to both your
questions is yes for n a power of 4.

Note that any such (directed) graph above will be a
regular graph on n vertices with degree ceil(n/2), or
floor(n/2), depending on the symbol set.  The literature
on such adjacency matrices and graphs is extensive; you
might look at existing constructions and see which have
or preserve the sudoku relation, which is having the
right number and position of 1's in certain submatrices.