This question in stackExchange remained unanswered. 


Let $\mathbb F$ be a finite field. Denote by $M\_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$ . For a matrix $A∈M\_n(\mathbb F)$ what is the cardinality of $C\_{M\_n(\mathbb F)} (A)$ , the centralizer of $A$ in $M\_n(\mathbb F)$? There are papers about it? 



Centralizer of a Matrix over a Finite Field
Let me add some cases in which one has a clear answer:


[R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991., Corollary 4.4.18]. Let $F$ be a field and $n$ is a natural number. If
$A\in M\_n(F)$ is a cyclic matrix, then $C\_{M\_n(F)}(A)$ is the set of all matrices which are
polynomial in $A$ with coefficients in $F$.


Recall that a cyclic matrix in $M\_n(F)$ is a matrix whose minimal and characteristic polynomials are the same.


[Lemma 3 of S. Akbari et al. / Linear Algebra and its Applications 390 (2004) 345–355] Let $F$ be a field and $n\geq 2$. If $A$ is a non-scalar matrix in $M\_n(F)$ and
$C\_{M\_n(F)}(A)$ has maximum dimension over $F$, then $\dim\_F C\_{M\_n(F)}(A) = n^2 − 2n + 2$
and $A$ is similar to either $aI\_1 \oplus bI\_{n−1}$ or $aI\_n + bE\_{12}$, for some $a, b \in F$. 


See for the notation the latter mentioned paper.


I suggest you to look for papers on the commuting graphs of rings, you may find some other cases which are treated in the proofs. 
One paper is quoted above and the another is
S. Akbari, P. Raja / Linear Algebra and its Applications 416 (2006) 1038–1047


