  In this paper, we study the combinatorics of a subcomplex of the Bloch-Kriz
cycle complex [4] used to construct the category of mixed Tate motives. The
algebraic cycles we consider properly contain the subalgebra of cycles that
correspond to multiple logarithms (as defined in [12]). We associate an algebra
of graphs to our subalgebra of algebraic cycles. We give a purely graphical
criterion for admissibilty. We show that sums of bivalent graphs correspond to
coboundary elements of the algebraic cycle complex. Finally, we compute the
Hodge realization for an infinite family of algebraic cycles represented by
sums of graphs that are not describable in the combinatorial language of [12].
