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

TL;DR

This paper explores the structure of adjacency matrices of finite graphs over prime fields, providing a complete classification of their direct sum decompositions using quadratic residues.

Contribution

It introduces a method to decompose adjacency matrices over finite prime fields, utilizing elementary transformations and quadratic residues, for any odd prime p.

Findings

Complete description of adjacency matrix decompositions over $\\mathbb{F}_p$ for odd primes

Application of elementary transformations to achieve direct sum decompositions

Use of quadratic residues to classify matrix structures

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 $\mathbb{F}_p$.