Vector spaces and Grassmann graphs over residue class rings

Abstract

Let ${\mathbb{Z}}_{p^s}$ be the residue class ring of integers modulo $p^s$, where $p$ is a prime number and $s$ is a positive integer. Using matrix representation and the inner rank of a matrix, we study the intersection, join, dimension formula and dual subspaces on vector subspaces of ${\mathbb{Z}}_{p^s}^n$. Based on these results, we investigate the Grassmann graph $G_{p^s}(n,m)$ over ${\mathbb{Z}}_{p^s}$. $G_{p^s}(n,m)$ is a connected vertex-transitive graph, and we determine its valency, clique number and maximum cliques. Finally, we characterize the automorphisms of $G_{p^s}(n,m)$.

25 pages