Affirmative Results on a Conjecture on the Column Space of the Adjacency Matrix

TL;DR

This paper investigates the ACK conjecture related to the row space of a graph's adjacency matrix, providing a unified framework and new constructions that support the conjecture's validity across various graph classes.

Contribution

It introduces a unified framework for families of graphs satisfying the ACK conjecture and demonstrates that certain graph operations preserve this property.

Findings

Several classes of graphs satisfy the ACK conjecture.

New graph constructions adhere to the conjecture.

Graph operations can preserve the ACK property.

Abstract

The Akbari-Cameron-Khosrovshahi (ACK) conjecture, which appears to be unresolved, states that for any simple graph $G$ with at least one edge, there exists a nonzero {$\{0,1\}$}-vector in the row space of its adjacency matrix that is not a row of the matrix itself. In this talk, we present a unified framework that includes several families and operations of graphs that satisfy the ACK conjecture. Using these fundamental results, we introduce new graph constructions and demonstrate, through graph structural and linear algebraic arguments, that these constructions adhere to the conjecture. Further, we show that certain graph operations preserve the ACK property. These results collectively expand the known classes of graphs satisfying the conjecture and provide insight into its structural invariance under composition and extension.