The critical polynomial of a finite simple graph 
is defined using its adjacency matrix. 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 (1989) introduced the matrices 
in connection with finite Abelian groups presented by graph-based
matrices. For a connected
graph 
,
Wang and Lu (2026) study nonnegative values 
with integers 
. Their density result requires
to be a positive definite matrix if 
, or a positive
semidefinite matrix of rank 
if 
, with the torsion subgroup
of its cokernel a cyclic
group. The resulting set of values has natural
density 1 among the nonnegative integers (Wang and
Lu 2026, Theorem 5).
