Sparse matrix with only $-1$ outside the diagonal

Question

We consider a simplicial decomposition of the $n$-dimensional disk with $r$ vertices, each vertex $v_i$ is associated with a weight $w_i$ which is a positive rational number. We have a matrix equation for the weights $M [w_1,...,w_r]^T = \vec{v}$.

This system of equations satisfy the following conditions:

1. the matrix $M$ has positive integers $m_1,...,m_r$ in the diagonal and every row has $n+1$ other non-trivial entries whcih are $-1$'s.
2. If there is a $-1$ in the position $(i,j)$, then the vertices $v_i$ and $v_j$ are joined by a line in the simplicial decomposition.
3. The vector $\vec{v}$ has only entries $0$'s or $1'$. If $\vec{v}$ has a one in the position $j$, then $v_j$ lies in the boundary of the disk.

I am trying to undestand the behaviour of the set of values $\min(w_1,...,w_r)$, when we fix $n$ and let $r$ increase.

For instance, if $n=2$, then the above matrix just represents continued fraction, and $\min(w_1,..,w_r)$ is always less than $1$ independent of the values of $M$. I suspect this matrices generalize continued fractions, so I am wondering if they show up elsewhere in mathematics.