Erdős-Ko-Rado theorem and bilinear forms graphs for matrices over residue class rings

2020/02/10 by Jun Guo, Guo, Jun
Mathematics · Computer Science · #Finite Group Theory Research #Limits and Structures in Graph Theory #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.2002.03560

Abstract

Let h=∏_i=1^tp_i^s_i be its decomposition into a product of powers of distinct primes, and ℤ_h be the residue class ring modulo h. Let 1≤ r≤ m≤ n and ℤ_h^{m× n} be the set of all m× n matrices over ℤ_h. The generalized bilinear forms graph over ℤ_h, denoted by \hboxBil_r(ℤ_h^{m× n}), has the vertex set ℤ_h^{m× 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 \hboxBil_r(ℤ_h^{m× n}). As a result, the Erdős-Ko-Rado theorem for ℤ_h^{m× n} is obtained.

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