There is a large literature devoted to studying the inverses of diagonally dominant matrices. I'd like to know if there is information about a so-to-say opposite situation: we have a matrix $A$ and want to find conditions under which $A^{-1}$ will be diagonally dominant.

One case which has been studied is that of strictly ultrametric matrices whose inverses are known to be Stieltjes matrices. But what if the original matrix has some negative entries?

UPDT: References to two papers, as requested some time ago:

(1) Li, Yaotang; Liu, Xin; Yang, Xiaoying; Li, Chaoqian Some new lower bounds for the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse. (English) [J] Electron. J. Linear Algebra 22, 630-643, electronic only (2011).

(2) Ostrowski, A.M. Note on bounds for determinants with dominant principal diagonal. (English) [J] Proc. Am. Math. Soc. 3, 26-30 (1952).