Adjacency matrices over a finite prime field and their direct sum decompositions

Abstract

In this paper, we discuss the adjacency matrices of finite undirected simple graphs over a finite prime field ${\mathbb{F}}_p$. We apply symmetric (row and column) elementary transformations to the adjacency matrix over ${\mathbb{F}}_p$ in order to get a direct sum decomposition by other adjacency matrices. In this paper, we give a complete description of the direct sum decomposition of the adjacency matrix of any graph over ${\mathbb{F}}_p$ for any odd prime $p$. Our key tool is quadratic residues of ${\mathbb{F}}_p$.

17 pages, 2 figures, 3 tables. Comments are welcome