Suppose that the matrix $M$ has non-positive off-diagonal elements. The matrix $M$ is
said to be an M-matrix if $M$ is non-singular and each entry of $C:=M^{−1}$
is non-negative.


I think I've proved the following implication: If $M$ is *symmetric*, then


$$ c\_{ij}=0 \Longrightarrow m\_{ij}=0 \quad (\textrm{for all } i\neq j)$$


Is this implication well-known? I did a quick literature search, but I could not find this precise statement. 



zero patterns of M-matrices and their invereses
I haven't encountered this exact result, but a closely related one is a complete characterization of the possible zero patterns for inverse M-matrices in <https://doi.org/10.1016/j.laa.2009.03.022>.


It seems to me that this one can be proved with the same techniques (decomposition into irreducible blocks + keeping track of nonzeros in the Neumann series). I like this result and the short formulation that you found, but probably it is not deep enough to warrant a publication alone.


Incidentally, it seems that the possible zero patterns for $C$ admit a simple description in the symmetric case: any M-matrix $M$ can be written (by permuting rows and columns conformably) as a block upper triangular matrix whose diagonal blocks are irreducible M-matrices; in the case in which $M$ is symmetric, this means a block diagonal matrix. The inverses of irreducible M-matrices are strictly positive, hence $M^{-1}$ is a block diagonal matrix with the same block sizes and only nonzeros in its diagonal blocks. So, in conclusion, $M^{-1}$ is a block diagonal matrix in which the diagonal blocks have all strictly positive entries.


