Automorphisms Of Generalized Orthogonal Graphs Of Odd Characteristic

The geometry of classical groups over finite fields has a wide range of applications in graph theory.Gu Zhenhua and Wan Zhexian had studied the properties of orthogonal graphs of odd characteristic and their automorphisms.On the base of their work,in this paper we construct a new family of graphs,called generalized orthogonal graphsΓGO2v+δ(q,m,S) over a finite field Fq of odd characteristic respect to S2v+δ,△by using all the m-dimensional totally isotropic spaces.Then we study some properties of them,and more important the groups of graph auto-morphisms are determined.Firstly,we can easily obtain that the generalized orthogonal graph is a distance regular graph with N(m,0,0; 2v+δ,△) vertices and the degree of it is q2(v-m)+δ(qm-1)/q-1.And the distance formula between any two vertices X and Y in the graph is (?)(r,t)(X, Y)= 2m-2t-r(r+t≤m).The diameter of it is d=min{2m,v}.Secondly, the geometry structure ofΓ(1)(M),Γ(2)(M) and the relationship between them are given.In addition,we also study the two important kinds of structures of the generalized orthogonal graph,namely maximal set and quasi-tetrahedron.Finally,on the base of these prepared work we gain the principal conclusion of this paper:Let 1