A det-extremal bipartite graph is a bipartite graph with two equally sized vertex parts whose biadjacency
matrix 
satisfies
 |
where 
and 
denote the determinant and permanent,
respectively. Equivalently, all nonzero terms associated with perfect
matchings have the same value of the permutation
symbol. A bipartite graph with equally sized
parts and no perfect matching satisfies the equality
trivially, with both sides zero (Funk et al. 2003, pp. 50-51).
Another characterization is that every graph cycle 
for which deleting all vertices of 
leaves a perfect matching
has length congruent to 2 modulo 4 (Funk et al. 2003, p. 51).
For a connected plane graph that is bipartite, has at least four vertices, and has every graph edge in a perfect matching, Li (2026, p. 6) proved that det-extremality is equivalent to the existence of a plane bipartite ear decomposition in which every interior graph face has boundary length congruent to 2 modulo 4.
For example, the cycle graph 
is det-extremal, whereas 
is not. Their biadjacency
matrices have permanents 2 in both cases, but determinants of absolute value 2 and 0, respectively.
