Erd\H{o}s-Ko-Rado theorem and bilinear forms graphs for matrices over residue class rings

TL;DR

This paper extends the Erdős-Ko-Rado theorem to matrices over residue class rings by analyzing the structure of bilinear forms graphs and determining maximum cliques.

Contribution

It introduces a generalized bilinear forms graph over residue class rings and characterizes its maximum cliques, leading to an Erdős-Ko-Rado type theorem in this setting.

Findings

Determined the clique number of the bilinear forms graph over residue class rings.

Characterized the geometric structure of maximum cliques.

Established an Erdős-Ko-Rado theorem for matrices over residue class rings.

Abstract

Let $h=\prod_{i=1}^{t}p_i^{s_i}$ be its decomposition into a product of powers of distinct primes, and $\mathbb{Z}_{h}$ be the residue class ring modulo $h$. Let $1\leq r\leq m\leq n$ and $\mathbb{Z}_{h}^{m\times n}$ be the set of all $m\times n$ matrices over $\mathbb{Z}_{h}$. The generalized bilinear forms graph over $\mathbb{Z}_{h}$, denoted by $\hbox{Bil}_r(\mathbb{Z}_{h}^{m\times n})$, has the vertex set $\mathbb{Z}_{h}^{m\times n}$, and two distinct vertices $A$ and $B$ are adjacent if the inner rank of $A-B$ is less than or equal to $r$. In this paper, we determine the clique number and geometric structures of maximum cliques of $\hbox{Bil}_r(\mathbb{Z}_{h}^{m\times n})$. As a result, the Erd\H{o}s-Ko-Rado theorem for $\mathbb{Z}_h^{m\times n}$ is obtained.