Smallest non-trivial conjugacy classes in simple groups and classes of involutions I am interested in finding the size of the smallest non-trivial conjugacy class
of the simple groups $PSL(d,q)$ with $d>2$, $Sz(q)$ with $q>2$ and $R(q)$ with $q>3$. 
My first question is related to the involutions in Suzuki and Ree groups. For $q=2^{2n+1}>2$, the Suzuki group $Sz(q)$ has a unique class of involutions of size $(q^2+1)(q-1)$. For $q=3^{2n+1}>3$, the Ree group $R(q)$ has a unique
class of involutions of size $q(q-1)(q+1)$. 

Are these classes the smallest non-trivial conjugacy classes?

With respect to the groups $PSL(d,q)$ my questions are the following:

Let $G=PSL(d,q)$ with $d>2$. Is some class of involutions the smallest non-trivial conjugacy class of $G$? 

I did some experiments with GAP and it seems that the smallest conjugacy class of $PSL(d,q)$, $Sz(q)$, $R(q)$, is some conjugacy class of involutions. 
 A: The answer for your second question is NO in general. Earlier Shawn Burkett and I did some work on the conjugacy classes of small size in the linear and unitary groups, see [J. Group Theory 16 (2013), 851-874.] Among other results, we showed that if $d\geq 6$ then the smallest non-trivial conjugacy class of $PSL(d,q)$ has size $\frac{(q^{n-1}-1)(q^n-1)}{q-1}$ and consists of certain unipotent elements. The order of these unipotent elements is the defined characteristic of the group. Hence these elements are involutions if and only if $q$ is even. When $n$ is smaller there are some exceptions and there could be a class of involutions of smallest size, but in general the answer is NO. 
As an example please look at $PSL(3,5)$ (though $3<6$ but the above result holds in this case!). The smallest non-trivial size of a conjugacy class of $PSL(3,5)$ is 744 and this smallest class consists of elements of order 5.
A: Concerning your first question for the Suzuki groups in characteristic 2, it's helpful to go back to the original announcement by Suzuki:  A new type of simple groups of finite order, Proc. Nat. Acad. Sci. USA 46 (1960), no. 6, 868-870 (available through JSTOR and maybe other channels).   Note that the order of such a group is $q^2(q-1)(q^2+1)$, with $q= 2^{2n+1}$.    Suzuki approached his groups in the setting of 2-transitive permutation groups, but later it was seen that they could be viewed efficiently as groups of Lie type: here one gets a split BN-pair of rank 1 and nice relations with an ambient finite group of type $B_2 = C_2$.   
Anyway, Suzuki realized that his groups are simple for $q>2$ and that each has only a few conjugacy classes of maximal (proper) subgroups.   You've pointed to the Sylow 2-subgroups of order $q^2$, each of which is the full centralizer of a nontrivial central involution (a unipotent element in Lie theory).    In addition there are other types of centralizers, of orders $q-1$ (split "torus"), $q+2r+1$, and $q-2r+1$ with $r=2^n$.  Here all centralizers turn out to be nilpotent, which is not too surprising from the Lie-theoretic viewpoint since the BN-pair has rank 1.
Given this much information, it's true that "small" classes (which have "big" centralizers) must consist of involutions here.  
In the case of the Ree groups in characteristic 3, contained in groups of Lie type $G_2$, the story is probably similar since you again have a "rank" 1 BN-pair (though I haven't checked all details).   Note that the Ree groups in characteristic 2 are more complicated, since they are twisted subgroups of Lie type groups $F_4$ and have "rank" 2. 
P.S. Looking at the order of a Ree group and some partial information about centralizers in Ward's 1966 paper here, I'm unsure where the data in the question about involutions fits in.   Anyway, there must be fairly complete data about centralizers in the literature.  ADDED: For a complete list of subgroups, see Theorem 6.5.5 in The Classification of the Finite Simple Groups, Number 3, by Gorenstein-Lyons-Solomon, AMS 1998.   These include all centralizers, and it appears that the largest subgroup in the list is the centralizer of (any) involution.   If so, this answers the question for Ree groups too.  (One of the two G-L-S sources is a paper by Kleidman here.)
