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

Abstract

Let be a finite field of q elements, an n-dimensional vector space over . 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 with vertex set , where is the set of all nonzero singular linear transformations over and is the set of all nontrivial subspaces of , and there is an undirected edge between and if and only if A maps W to the zero space, that is . 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 . In the present article, we solve this problem completely.