Orthogonal Systems in Finite Graphs

TL;DR

This paper develops an orthogonality theory for finite graphs to analyze the structure and automorphisms of partially commutative groups, also known as right-angled Artin groups, based on their graph representations.

Contribution

It introduces a new orthogonality framework for graphs that enhances understanding of the centraliser lattice and automorphism groups of partially commutative groups.

Findings

Provides tools for analyzing the structure of centraliser lattices.

Facilitates study of automorphism groups of right-angled Artin groups.

Establishes a foundation for further algebraic and combinatorial investigations.

Abstract

Given a finite graph G there is a corresponding group given by the presentation with generators the vertices of G and a relation [x,y]=1 for generators x and y precisely when (x,y) is an edge of G. Such groups are known as partially commutative groups (or right-angled Artin groups). In this paper we construct orthogonality theory for graphs with the study of partially commutative groups in mind. The theory developed here provides tools for the study of the structure of the centraliser lattice of partially commutative groups and for their automorphism groups.