Automorphisms and independent number of single nonzero component graph over a vector space

In this paper, we introduce a graph structure, called the single non-zero component graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="normal">Γ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">V</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> Γ(V) , on a finite dimensional vector space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="double-struck">V</m:mi> </m:math> V . We study the minimal domination sets as well as minimum independent domination sets, and compute the domination number of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="normal">Γ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">V</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> Γ(V) . Furthermore, using the domination number and minimum independent domination sets, respectively, we prove that two vector spaces are isomorphic if and only if the corresponding single nonzero component graphs are isomorphic, and determine the automorphisms of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="normal">Γ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">V</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> Γ(V) . Finally, we characterize the maximum independent sets when the dimension of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="double-struck">V</m:mi> </m:math> V is not more than 5.