ABSTRACT
Let R be a finite commutative ring with identity, and β a bilinear form on
. In this work, we count the numbers of free submodules and totally isotropic free submodules of
of rank s by using the lifting idea. We define the graph whose vertex set is the set of totally isotropic free submodules of
of rank s called the generalized bilinear form graph. We study this graph when
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. We complete our work by proving that we can decompose the graphs over a finite commutative ring into the tensor products of graphs over finite local rings.
Disclosure statement
No potential conflict of interest was reported by the authors.