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

#### Abstract

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.

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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