The paper presents new diagonal dominance type nonsingularity conditions for n × n matrices formulated in terms of circuits of length not exceeding a fixed number r ≥ 0 and simple paths of length r in the digraph of the matrix. These conditions are intermediate between the diagonal dominance conditions in terms of all paths of length r and Brualdi’s diagonal dominance conditions, involving all the circuits. For r = 0, the new conditions reduce to the standard row diagonal dominance conditions \(\left| {a_{ii} } \right| \geqslant \sum\limits_{j \ne i} {\left| {a_{ij} } \right|} \), i = 1, ..., n, whereas for r = n they coincide with the Brualdi circuit conditions. Thus, they connect the classical Levy-Desplanques theorem and the Brualdi theorem, yielding a family of sufficient nonsingularity conditions. Further, for irreducible matrices satisfying the new diagonal dominance conditions with nonstrict inequalities, the singularity/nonsingularity problem is solved. Also the new sufficient diagonal dominance conditions are extended to the so-called mixed conditions, simultaneously involving the deleted row and column sums of an arbitrary finite set of matrices diagonally conjugated to a given one, which, in the simplest nontrivial case, reduce to the old-known Ostrowski conditions \(\left| {a_{ii} } \right| > \left( {\sum\limits_{j \ne i} {\left| {a_{ij} } \right|} } \right)^\alpha \left( {\sum\limits_{j \ne i} {\left| {a_{ji} } \right|} } \right)^{1 - \alpha } \), i = 1, ..., n, 0 ≤ α ≤ 1. The nonsingularity conditions obtained are used to provide new eigenvalue inclusion sets, depending on r, which, as r varies from 0 to n, serve as a bridge connecting the union of Gerschgorin’s disks with the Brualdi inclusion set. Bibliography: 16 titles.