Maximum determinant and permanent of sparse 0-1 matrices

We prove that the maximum determinant of an n×n matrix, with entries in {0,1} and at most n+k non-zero entries, is 2k/3, which best possible when k a multiple 3. This result solves conjecture Bruhn Rautenbach. also obtain upper bound on number perfect matchings C4-free bipartite graphs based edges, which, sparse case, improves classical Bregman's inequality for permanents. tight, as equality achieved by graph formed vertex disjoint union 6-vertex cycles.