Generalized symplectic graphs and generalized orthogonal graphs over finite commutative rings

ABSTRACT

Let R be a finite commutative ring with identity, $n\in \mathbb{N}$ and β a bilinear form on $R^n$. In this work, we count the numbers of free submodules and totally isotropic free submodules of $R^n$ of rank s by using the lifting idea. We define the graph whose vertex set is the set of totally isotropic free submodules of $R^n$ of rank s called the generalized bilinear form graph. We study this graph when $(R^n,\beta )$ 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.

KEYWORDS:

• Generalized bilinear form graph
• local ring
• orthogonal graph
• symplectic graph

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 05C25
• 05C60

Disclosure statement

No potential conflict of interest was reported by the authors.