On the lower bound for diameter of commuting graph of prime-square sized matrices

David Dolžan, Damjana Kokol Bukovšek, Bojan Kuzma


It is known that the diameter of commuting graph of $n$-by-$n$ matrices is bounded above by six if the graph is connected.
In the commuting graph of $p^2$-by-$p^2$ matrices over a  sufficiently large field which admits a cyclic Galois extension of degree $p^2$ we construct  two matrices at distance
at least five. This shows that five is  the lower bound  for its diameter. Our results are applicable for all sufficiently large  finite fields as well as  for the  field of rational numbers.


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