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

S. Akansha, K. C. Sivakumar

TL;DR

The paper advances the ACK conjecture by developing a kernel-vector, zero-sum subset framework and applying it to construct broad families of graphs that satisfy ACK, including satellite graphs and various core-graph constructions. It proves that several natural graph operations, notably the $K_2$-cartesian product under spectral conditions and vertex-adding procedures, preserve the ACK property, and it shows how core graphs of arbitrary nullity can abide by ACK through matrix-structural criteria. The results collectively widen the classes of graphs for which ACK holds and argue against potential counterexamples within the class ${\cal C}$, offering constructive tools for building ACK-satisfying graphs and deepening understanding of the conjecture’s structural invariants.

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.

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

TL;DR

The paper advances the ACK conjecture by developing a kernel-vector, zero-sum subset framework and applying it to construct broad families of graphs that satisfy ACK, including satellite graphs and various core-graph constructions. It proves that several natural graph operations, notably the -cartesian product under spectral conditions and vertex-adding procedures, preserve the ACK property, and it shows how core graphs of arbitrary nullity can abide by ACK through matrix-structural criteria. The results collectively widen the classes of graphs for which ACK holds and argue against potential counterexamples within the class , offering constructive tools for building ACK-satisfying graphs and deepening understanding of the conjecture’s structural invariants.

Abstract

The Akbari-Cameron-Khosrovshahi (ACK) conjecture, which appears to be unresolved, states that for any simple graph with at least one edge, there exists a nonzero {}-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.
Paper Structure (7 sections, 17 theorems, 33 equations, 8 figures)

This paper contains 7 sections, 17 theorems, 33 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A reformulation of the conjecture
  4. Graphs in C satisfying ACK conjecture
  5. Satellite graphs
  6. Two graph operations and the ACK conjecture
  7. Core graphs satisfying ACK conjecture

Key Result

Theorem 1

sciriha2025potential Let $G$ be a singular base graph and $v$ be a non-duplicated vertex added to $G$ such that $v$ is adjacent to at least one vertex in $G$. Then $v$ is not a Parter vertex of $G+v$ iff $a^v \in R(A_G)$.

Figures (8)

  • Figure 3: The degree sequence is $\{2,2,2,2, 4,4,4,4,4,4,4, 5,6, 13\}$.
  • Figure 4: Satellite graphs for $k=3,4$ and $5$.
  • Figure 5: Graph $E_{2k}$ for $k =4,5$ and $6$.
  • Figure 6: The first two graphs have degree sequence $\{2,2,2,3,3,4,4,4,4,4,4,4,4,5,6,15\}$ with dependence relation $a^7+a^9+a^{12}+a^{14}-a^4-a^{11}-2a^{15}=0$. The next two graphs have degree sequences $\{2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5,6,17\}$ and $\{2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,5,6,17\}$, with dependence relations $a^3-a^5+a^{15}-a^{13}=0$ and $a^4-a^5+a^6-a^8+a^9-2a^3+a^{12}-a^{14}-a^{15}-a^{16}+2a^{18}=0$, respectively.
  • Figure 7: Graph $G = K_2 \square H$
  • ...and 3 more figures

Theorems & Definitions (48)

  • Conjecture 1.1
  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • Definition 3.1: Zero-sum subset
  • Lemma 2
  • proof
  • Proposition 3.1
  • proof
  • ...and 38 more