  In [15] and [16], we constructed a graph W*-probability space over its
diagonal subalgebra. To study this strudture is a good example of Speicher's
amalgamated W*-probability spaces. In this paper, we will observe the diagonal
compressed random variables. we will compute the moments and cumulants of them
and we will show that the freeness on diagonal compressed random variables in
the graph W*-probability space is preserved by that of (uncompressed) random
variables.
