  Graph labeling is a well-known and intensively investigated problem in graph
theory. Sparse anti-magic squares are useful in constructing vertex-magic
labeling for graphs. For positive integers $n,d$ and $d<n$, an $n\times n$
array $A$ based on $\{0,1,\cdots,nd\}$ is called \emph{a sparse anti-magic
square of order $n$ with density $d$}, denoted by SAMS$(n,d)$, if each element
of $\{1,2,\cdots,nd\}$ occurs exactly one entry of $A$, and its row-sums,
column-sums and two main diagonal sums constitute a set of $2n+2$ consecutive
integers. An SAMS$(n,d)$ is called \emph{regular} if there are exactly $d$
positive entries in each row, each column and each main diagonal. In this
paper, we investigate the existence of regular sparse anti-magic squares of
order $n\equiv1,5\pmod 6$, and it is proved that for any $n\equiv1,5\pmod 6$,
there exists a regular SAMS$(n,d)$ if and only if $2\leq d\leq n-1$.
