Orthogonal Inner Product Graphs over Finite Fields of Odd Characteristic

Abstract

Let Fq be a finite field of odd characteristic and 2ν+δ≥2 be an integer with δ=0,1, or 2. The orthogonal inner product graph Oi2ν+δ,q over Fq is defined, and a class of subgroup of the automorphism groups of Oi2ν+δ,q is determined. We show that Oi2ν+δ,q is a disconnected graph if 2ν+δ=2; otherwise, it is not. Moreover, we give necessary and sufficient conditions for two vertices and two edges of Oi2ν+δ,q, respectively, which are in the same orbit under the action of a subgroup of the automorphism group of Oi2ν+δ,q.