Sparse anti-magic squares are useful in constructing vertex-magic labelings for bipartite graphs. An n×n array based on {0,1,…,nd} is called a sparse anti-magic square of ordernwith densityd (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, we investigate the existence of regular sparse anti-magic squares with densities d=3,5 and it is proved that there exists a regular SAMS(n,3) if and only if n≥4 and there exists a regular SAMS(n,5) if and only if n≥6.
A magic square is an n×n array of numbers whose rows, columns, and the two diagonals sum to μ. A regular magic square satisfies the condition that the entries symmetrically placed with respect to the ...center sum to 2μn. Using circulant matrices we describe a construction of regular classical magic squares that are nonsingular for all odd orders. A similar construction is given that produces regular classical magic squares that are singular for odd composite orders. This paper is an extension of 3.
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