Automorphisms group and domination number of a graph based on linear transformations and their kernels

Let F-q be a finite field of q elements, V an n-dimensional vector space over F-q. Recently, Wang et al. in [Automorphisms and domination numbers of transformation graphs over vector spaces. Linear Multilinear Algebra. 2019;67:1350-1363] defined a graph Gamma'(V) with vertex set T boolean OR W, where 'T is the set of all nonzero singular linear transformations over V and W is the set of all nontrivial subspaces of V, and there is an undirected edge between A is an element of T and W is an element of W if and only if A maps W to the zero space, that is A(W) = {0}. In the end of Wang et al. [Automorphisms and domination numbers of transformation graphs over vector spaces. Linear Multilinear Algebra. 2019;67:1350-1363], the authors posed an open problem: Characterizing the domination number and the automorphisms of Gamma'(V). In the present article, we solve this problem completely.