Geodesic
The singularity/nonsingularity problem for matrices satisfying diagonal dominance conditions in terms of directed graphs
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVII, Tome 309 (2004), pp. 40-83

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The paper considers the singularity/nonsingularity problem for matrices satisfying certain conditions of diagonal dominance. The conditions considered extend the classical diagonal dominance conditions and involve the directed graph of the matrix in question. Furthermore, in the case of the so-called mixed diagonal dominance, the corresponding conditions are allowed to involve both row and column sums for an arbitrary finite set of matrices diagonally conjugated to the original matrix. Conditions sufficient for the nonsingularity of quasi-irreducible matrices strictly diagonally dominant in certain senses are established, as well as necessary and sufficient conditions of singularity/nonsingularity for weakly diagonally dominant matrices in the irreducible case. The results obtained are used to describe inclusion regions for eigenvalues of arbitrary matrices. In particular, a direct extension of the Gerschgorin ($r=1$) and Ostrowski–Brauer ($r=2$) theorems to $r\geqslant3$ is presented. 
@article{ZNSL_2004_309_a4,
     author = {L. Yu. Kolotilina},
     title = {The singularity/nonsingularity problem for matrices satisfying  diagonal dominance conditions in terms of directed graphs},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {40--83},
     publisher = {mathdoc},
     volume = {309},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_309_a4/}
}
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L. Yu. Kolotilina. The singularity/nonsingularity problem for matrices satisfying  diagonal dominance conditions in terms of directed graphs. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVII, Tome 309 (2004), pp. 40-83. http://geodesic.mathdoc.fr/item/ZNSL_2004_309_a4/