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

TL;DR

This work resolves the Akbari–Cameron–Khosrovshahi conjecture for all graphs with diameter at least $4$ by constructing a non-zero $(0,1)$-vector in the row space of the adjacency matrix that is not a row, notably via the vector $R_2+R_5$ on a shortest diameter-$4$ path. It also provides partial validation for diameter $2$ or $3$ cases, including full verification for graphs with $r(\Gamma)\le 5$ and the introduction of infinite graph families $\mathcal{H}$ (diameter $2$) and $\mathcal{D}$ (diameter $2$–$3$) closed under degree-$2$ vertex duplication and vertex multiplication, with the conjecture holding in these classes. The approach hinges on the vertex-multiplication operation $\Gamma \odot m$, which preserves rank and allows lifting of $(0,1)$-row-space vectors from base graphs to larger constructions, tying the conjecture to diameter, rank, and graph-dilation properties. Collectively, the results advance understanding of the interplay between adjacency-row spaces and graph structure, offering a path toward a general resolution of the conjecture and informing related questions on graph rank and spectrum.

Abstract

We look for a non-zero $(0, 1)$-vector in the row space of the adjacency matrix $A(Γ)$ of a graph $Γ,$ provided $Γ$ has at least one edge. Akbari, Cameron, and Khosrovshahi conjectured that there exists a non-zero $(0,1)$-vector in the row space of $A(Γ)$ (over the real numbers) which does not occur as a row of $A(Γ).$ 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.

Figures (7)

• Figure 1: An illustration of the conjecture
• Figure 2: Explanation of multiplication of vertices
• Figure 3: $\Gamma_1=C_5, \Gamma_2$ is the graph obtained from $C_5$ by duplicating three degree-$2$ vertices, and $\Gamma_3$ is obtained from $C_5$ by duplicating six degree-$2$ vertices. Similarly, the graph $\Gamma_4$ is obtained from $\Gamma_0$ by duplicating six degree-$2$ vertices and $\Gamma_5$ is the Petersen graph.
• Figure 4: Reduced graphs of rank $4$
• Figure 5: Reduced graphs of rank $5$

Theorems & Definitions (10)

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• proof : Proof of Theorem \ref{['thm: proof of conj diam geq 4']}
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• proof : Proof of Theorem \ref{['thm: rank 5 graph follows conj']}
• proof : Proof of Theorem \ref{['thm: dim 2 graph m=2n-5']}