Determine the exact domination number of singular difference graphs

Determine the exact domination number of the singular difference graph of the matrix space M_n(F_q) over a finite field F_q, thereby establishing whether the upper bound γ(Γ) ≤ n(q−1)+1 supplied by the explicit dominating set is optimal in general.

Background

The paper studies the singular difference graph Γ of M_n(F_q), whose vertices are n×n matrices over F_q and in which two distinct matrices are adjacent exactly when their difference is singular. For this graph, the paper constructs the dominating set D = {cE_{nj} : c ∈ F_q, 1 ≤ j ≤ n}, together with the zero matrix, and obtains the upper bound γ(Γ) ≤ n(q−1)+1.

The construction does not prove that this bound is minimal. The authors explain that proving optimality would require a matching lower bound showing that every dominating set has at least n(q−1)+1 vertices, and note that standard graph-theoretic lower bounds do not provide this. They verify equality computationally for M_2(F_2) and M_2(F_3), but leave the general case unresolved.