For a graph 
on vertices 
, ..., 
with adjacency matrix 
, define
 |
(1)
|
The critical polynomial of 
is the multivariate
polynomial obtained as the determinant
 |
(2)
|
Setting every variable equal gives
 |
(3)
|
where 
is the identity matrix, so the usual characteristic
polynomial is a diagonal specialization of the critical polynomial.
Lorenzini introduced the matrices 
in connection with finite Abelian
groups presented by graph-based matrices.
Wang and Lu (2026) prove a density result for nonnegative values of 
obtained from integers 
when the resulting matrices
are positive semidefinite matrices
and satisfy cyclic-cokernel conditions.
