Regular sparse anti-magic squares with maximum density

Sparse anti-magic squares are useful in constructing vertex-magic labelings for bipartite graphs. An n x n array based on {0,1,..., nd} is called a sparse anti-magic square of order n with density d (d < n), denoted by SAMS(n, d), if its row-sums, column-sums and two main diagonal sums constitute a set of 2n + 2 consecutive integers. A SAMS(n, d) is called regular if there are d positive entries in each row, each column and each main diagonal. In this paper, some constructions of regular sparse anti-magic squares are provided and it is shown that there exists a regular SAMS(n, n-1) if and only if n >= 4.