What can be said about the adjacency algebra (or coherent algebra) of an asymmetric graph? Is it always $M_n(\mathbb{C})$? If not, what's a counterexample?
2026-03-25 07:42:59.1774424579
Adjacency algebra of asymmetric graph
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There are asymmetric strongly regular graphs on 25 vertices, and larger, and the adjacency algebra of an srg has dimension three.
The adjacency algebra of a graph on $n$ vertices has dimension at most $n$, use Cayley-Hamilton to show that if $k\ge0$, then $A^n$ is a linear combination of the first $n$ powers of $A$. So trivial cases aside you never get the full matrix algebra.