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 Fp. We apply symmetric (row and column) elementary transformations to the adjacency matrix over Fp 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 Fp for any odd prime p. Our key tool is quadratic residues of Fp.