A new notion of vertex independence and rank for finite graphs

TL;DR

This paper introduces a novel concept of vertex independence and rank in finite graphs based on boolean independence of matrix columns, linking geometric properties to these new measures.

Contribution

It proposes a new boolean-based vertex independence and rank, along with foundational properties and theorems connecting geometric aspects of graphs.

Findings

Defines boolean independence of columns in a graph-associated matrix

Establishes basic properties of the new independence and rank

Relates geometric properties of graphs to the introduced measures

Abstract

A new notion of vertex independence and rank for a finite graph G is introduced. The independence of vertices is based on the boolean independence of columns of a natural boolean matrix associated to G. Rank is the cardinality of the largest set of independent columns. Some basic properties and some more advanced theorems are proved. Geometric properties of the graph are related to its rank and independent sets.