The commuting graph of a noncommutative ring 
is the simple graph whose
vertices are the elements outside the ring
center, with two distinct vertices adjacent iff the elements commute, 
(Akbari et al. 2004). For a matrix
ring, adjacency is the relation of being commuting
matrices.
Shahverdi and Shahverdi (2026) proved that if a ring 
with multiplicative
identity has a commuting graph isomorphic to that of 
for any 
and any finite field 
, then 
is isomorphic to that matrix ring.
Thus the commuting graph determines these finite matrix
rings of even size. Shahverdi and Shahverdi (2026) report AI assistance with
checking the proof.
