The automorphism group and fixing number of orthogonality graph over a vector space

Abstract

Let [Formula: see text] be a field, and [Formula: see text] the [Formula: see text]-dimensional row vector space over [Formula: see text]. The orthogonality graph [Formula: see text] of [Formula: see text] is an undirected simple graph which has [Formula: see text] as its vertex set, and for distinct [Formula: see text], [Formula: see text] if and only if [Formula: see text], where [Formula: see text] is the transpose of [Formula: see text]. When [Formula: see text] is finite, it is shown that any automorphism of [Formula: see text] can be decomposed into the product of a row-orthogonal automorphism and either a permutation automorphism or a field automorphism; moreover, the fixing number and metric dimension of [Formula: see text] are considered.