Summary: Computing q(Km,n)
Let G be a bipartite graph with parts (X, Y ) such that 0 < |X| = m n = |Y |. Define B(G) to be
the set of all real m × n matrices B whose rows and columns are indexed by X and Y , respectively and
for which Bx,y = 0 if and only if x y. We have the following:
Theorem 1. For any non-empty bipartite graph G, if there is a B B(G) whose set of rows and set of
columns are orthonormal, then q(G) = 2.
Proof. Assume that there is such a B B(G). Then since the row and the column spaces of B have the
same dimension, we must have m = n. Also, the following matrix is in S(G):
But we have
This implies that A has exactly 2 distinct eigenvalues. Therefore q(G) = 2.
We also observe the following: