Maybe as a follow-up to my last question : What can be said about the coherent algebra of an asymmetric graph? I.e. the smallest unital *-subalgebra of $M_n(\mathbb{C})$ closed under Schur (entrywise) products and containing the all-ones-matrix and the adjacency matrix of an undirected graph with trivial automorphism group. Is there any easy description of it?
2026-03-25 03:03:54.1774407834
What can be said about the coherent algebra of an asymmetric graph?
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The coherent algebra of a strongly regular graph is equal to its adjacency algebra, and so is commutative of dimension three.
At the other extreme, O’Rourke and Touri have prove that almost all graphs are controllable, equivalently the adjacency matrix and the all-ones matrix together generate the full matrix algebra, whence the coherent algebra is the full matrix algebra. Note that controllable graphs are asymmetric.