Erdős-Ko-Rado theorem and bilinear forms graphs 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 $1\le r\le m\le n$ 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 ${Bil}_r({\mathbb{Z}}_h^{m\times n})$, 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 ${Bil}_r({\mathbb{Z}}_h^{m\times n})$. As a result, the Erdős-Ko-Rado theorem for ${\mathbb{Z}}_h^{m\times n}$ is obtained.

13 pages