[Paper Review] Automorphism related parameters of a graph associated to a finite vector space

Hira Benish, Imran Javaid+1 more|arXiv (Cornell University)|Apr 25, 2018

Graph Labeling and Dimension Problems12 references3 citations

TL;DR

This paper investigates automorphism-related parameters of the non-zero component graph associated with a finite-dimensional vector space over a finite field. It introduces the fixing neighborhood of vertex pairs and determines the fixed number—the minimum size of a set that breaks all nontrivial automorphisms—showing that for vector spaces over F₂, the fixed number is 2ⁿ⁻¹, and over F_q with q ≥ 3, it is |G(𝕍)| − 1.

ABSTRACT

In this paper, we discuss automorphism related parameters of a graph associated to a finite vector space. The fixing neighborhood of a pair $(u,v)$ of vertices of a graph $G$ is the set of all those vertices $w$ of $G$, such that the orbits of $u$ and $v$ under the action of stabilizer of $w$ are not equal. The fixed number of a graph is the minimum number $k$ such that every subset of vertices of $G$ of cardinality $k$ is a fixing set of $G$. We study some properties of automorphisms of a graph associated to finite vector space and find the fixing neighborhood of pair of vertices of the graph. We also find the fixed number of the graph. It is shown that, for every positive integer $N$, there exists a graph $G$ with $fxd(G)-fix(G)\geq N$, where $fxd(G)$ is the fixed number and $fix(G)$ is the fixing number of $G$.

Motivation & Objective

• To analyze automorphisms of the non-zero component graph of a finite vector space.
• To define and compute the fixing neighborhood of vertex pairs in this graph.
• To determine the fixed number—the smallest size of a vertex set that destroys all nontrivial automorphisms.
• To establish conditions under which the fixed number differs significantly from the fixing number.
• To explore structural properties of the graph linked to vector space symmetries and field characteristics.

Proposed method

• The non-zero component graph is constructed with non-zero vectors as vertices, where two vectors are adjacent if they share a basis vector with non-zero coefficient.
• The fixing neighborhood of a vertex pair (u,v) is defined as the set of vertices w such that the stabilizer of w does not preserve the orbit of u and v under automorphism.
• The fixed number is computed using the characterization that fxd(G) = r + 1, where r is the largest size of a non-fixing set.
• For F₂, a non-fixing set A of size 2ⁿ⁻¹ − 1 is constructed by grouping vectors based on shared or absent basis vectors bₗ and bₘ.
• For F_q with q ≥ 3, twin classes of vertices with identical neighborhoods are identified, leading to the result that fxd(G) = |G| − 1.
• Theoretical results from graph automorphism theory and vector space structure are combined to prove that any superset of the constructed non-fixing set becomes a fixing set.

Experimental results

Research questions

• RQ1 What is the fixed number of the non-zero component graph of a finite-dimensional vector space over F₂?
• RQ2 How does the fixed number relate to the structure of automorphisms in the graph?
• RQ3 Can the fixed number exceed the fixing number by an arbitrary amount, and if so, under what conditions?
• RQ4 What role do twin vertices—vertices with identical neighborhoods—play in determining the fixed number?
• RQ5 How does the field size (q) affect the fixed number of the non-zero component graph?

Key findings

• For a finite vector space of dimension n ≥ 3 over F₂, the fixed number of its non-zero component graph is exactly 2ⁿ⁻¹.
• The largest non-fixing set in the non-zero component graph over F₂ has size 2ⁿ⁻¹ − 1, which directly determines the fixed number via fxd(G) = r + 1.
• Over F_q with q ≥ 3, the fixed number is |G(𝕍)| − 1, due to the presence of twin vertex classes with identical neighborhoods.
• The fixed number can exceed the fixing number by an arbitrarily large amount N, as shown by constructing graphs where fxd(G) − fix(G) ≥ N for any positive integer N.
• The fixing neighborhood of a vertex pair (u,v) contains all vertices w for which the stabilizer of w does not preserve the orbit of u and v, and this concept is essential in proving the fixed number.
• The automorphism group of the non-zero component graph maps basis vectors to signed permutations of basis vectors, preserving the graph structure independently of the chosen basis.