Generalized symplectic graphs and generalized orthogonal graphs over finite commutative rings

Abstract

Let R be a finite commutative ring with identity, n ∈ N and β a bilinear form on Rⁿ. In this dissertation, we count the numbers of free submodules and totally isotropic free submodules of Rⁿ of rank s by using the lifting idea. We define the generalized bilinear form graph whose vertex set is the set of totally isotropic free submodules of Rⁿ of rank s and the adjacency condition is given by some rank conditions. We study this graph when (Rⁿ, β) is a symplectic space and an orthogonal space. We can determine the degree of each vertex of these graphs. If R is a finite local ring, we show that these graphs are arc transitive and obtain their automorphism groups. Finally, we prove that we can decompose the graphs over a finite commutative ring into the tensor products of graphs over finite local rings.