Component graphs of vector spaces and zero-divisor graphs of ordered sets

TL;DR

This paper explores the structure of component graphs of finite-dimensional vector spaces through zero-divisor graphs of lattices and rings, revealing their join structures and characterizing their perfect and chordal properties.

Contribution

It introduces a novel approach to analyze component graphs using zero-divisor graphs and establishes their join relations with Boolean algebras and complete graphs.

Findings

Component graphs are the join of zero-divisor graphs of Boolean algebras and complete graphs.

Characterization of perfect and chordal properties of these graphs.

New connections between vector space component graphs and algebraic zero-divisor graphs.

Abstract

In this paper, nonzero component graphs and nonzero component union graphs of finite dimensional vector space are studied using the zero-divisor graph of specially constructed 0-1-distributive lattice and the zero-divisor graph of rings. Further, we define an equivalence relation on nonzero component graphs and nonzero component union graphs to deduce that these graphs are the graph join of zero-divisor graphs of Boolean algebras and complete graphs. In the last section, we characterize the perfect and chordal nonzero component graphs and nonzero component union graphs.