Orthogonal Inner Product Graphs over Finite Fields of Odd Characteristic

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