# Vector spaces and Grassmann graphs over residue class rings

## TL;DR

This paper explores the structure of vector spaces over residue class rings, analyzing subspace interactions and automorphisms of Grassmann graphs over these rings, revealing their combinatorial properties.

## Contribution

It introduces a matrix-based approach to study subspaces over residue class rings and characterizes the automorphisms of Grassmann graphs in this setting.

## Key findings

- Grassmann graph $G_{p^s}(n,m)$ is connected and vertex-transitive
- Valency, clique number, and maximum cliques are determined
- Automorphisms of the Grassmann graph are characterized

## 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 $\mathbb{Z}^n_{p^s}$. 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 $G_{p^s}(n,m)$.