Existence of a Non-Zero $(0,1)$-Vector in the Row Space of Adjacency Matrices of Simple Graphs

TL;DR

This paper proves a conjecture that graphs with diameter at least 4 have a non-zero (0,1)-vector in their adjacency matrix row space not equal to any row, advancing understanding of graph adjacency properties.

Contribution

The paper confirms the conjecture for graphs with diameter ≥ 4 and provides partial progress for diameter 2 or 3 cases.

Findings

Confirmed the conjecture for graphs with diameter ≥ 4

Progress made on the conjecture for diameter 2 and 3 graphs

Enhanced understanding of adjacency matrix row space in graph theory

Abstract

We look for a non-zero $(0, 1)$-vector in the row space of the adjacency matrix $A(\Gamma)$ of a graph $\Gamma,$ provided $\Gamma$ has at least one edge. Akbari, Cameron, and Khosrovshahi conjectured that there exists a non-zero $(0,1)$-vector in the row space of $A(\Gamma)$ (over the real numbers) which does not occur as a row of $A(\Gamma).$ This conjecture can be easily verified for graphs having diameter is equal to $1$ (complete graphs). In this article, we affirmatively prove this conjecture for any graph whose diameter is $\geq 4.$ Furthermore, in the remaining two cases that is, for graphs with diameter is equal to $2$ or $3,$ we report some progress in support of the conjecture.