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Spectral Bounds of the Generating Graph of $\mathbb{Z}_n.$

Kavita Samant, A. Satyanarayana Reddy

TL;DR

The paper addresses spectral bounds for the generating graph of the cyclic group, $Z_n$, by leveraging an equitable partition and an $H$-join decomposition to relate $E_n$ to the square-free case $E_{n_0}$. The adjacency spectrum is analyzed via a symmetric matrix $Q_n$, with a key reduction $\tilde{Q}_n=(n/n_0)\tilde{Q}_{n_0}$ that allows eigenvalue bounds to be derived and tied to the simpler case. For the Laplacian, a generalized join framework yields an explicit spectrum consisting of component eigenvalues and a central matrix $L_Q$, which similarly scales with $n/n_0$, reducing the problem to $n_0$. The work also characterizes minimal generating sets of $Z_n$ and provides structural graph properties, showing that these co-maximal graphs are amenable to tight, scalable spectral analysis across all $n$ with the same prime support.

Abstract

Let $G$ be a group. A group is said to be $k$-generated if it can be generated by its $k$ elements. A generating set of $G$ is called a minimal generating set if no proper subset of it generates $G.$ A minimal generating set of a group can have different sizes. The generating graph $Γ(G)$ of a group $G$ is defined as a graph with the vertex set $G$, where two distinct vertices are adjacent if they together generate $G.$ This graph is particularly useful when studying 2-generated groups. In this context, consider the group $G = \mathbb{Z}_n$, the integers modulo $n.$ In this paper, we explore various graph-theoretic properties of the generating graph $Γ(\mathbb{Z}_n)$ and investigate the spectra of its adjacency and Laplacian matrices. Additionally, we explicitly determine the set of all possible minimal generating sets of $\mathbb{Z}_n$ of size $k.$

Spectral Bounds of the Generating Graph of $\mathbb{Z}_n.$

TL;DR

The paper addresses spectral bounds for the generating graph of the cyclic group, , by leveraging an equitable partition and an -join decomposition to relate to the square-free case . The adjacency spectrum is analyzed via a symmetric matrix , with a key reduction that allows eigenvalue bounds to be derived and tied to the simpler case. For the Laplacian, a generalized join framework yields an explicit spectrum consisting of component eigenvalues and a central matrix , which similarly scales with , reducing the problem to . The work also characterizes minimal generating sets of and provides structural graph properties, showing that these co-maximal graphs are amenable to tight, scalable spectral analysis across all with the same prime support.

Abstract

Let be a group. A group is said to be -generated if it can be generated by its elements. A generating set of is called a minimal generating set if no proper subset of it generates A minimal generating set of a group can have different sizes. The generating graph of a group is defined as a graph with the vertex set , where two distinct vertices are adjacent if they together generate This graph is particularly useful when studying 2-generated groups. In this context, consider the group , the integers modulo In this paper, we explore various graph-theoretic properties of the generating graph and investigate the spectra of its adjacency and Laplacian matrices. Additionally, we explicitly determine the set of all possible minimal generating sets of of size
Paper Structure (10 sections, 25 theorems, 80 equations, 2 figures, 3 tables)

This paper contains 10 sections, 25 theorems, 80 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Minimal Generating Sets of ${\mathbb Z}_n$
  3. Generating Graph or Comaximal Graph of ${\mathbb Z}_n.$
  4. Graph Decomposition
  5. A Quick Analysis of the Graph Structure
  6. Adjacency Matrix Spectrum
  7. Preliminaries
  8. Eigenvalue Bounds
  9. Laplacian Matrix Spectrum
  10. Conclusion

Key Result

Theorem 2.0.1

For a given $n,$ the size of an equivalence class of the set $\{0,1,2,\dots,n-1\}$ is given by

Figures (2)

  • Figure 1: The generating graphs ${\mathcal{E}}_3,{\mathcal{E}}_6, {\mathcal{E}}_{8}\text{ and } {\mathcal{E}}_{12}.$
  • Figure 2: The graph ${\mathbb H}$ for ${\mathcal{E}}_{30}.$

Theorems & Definitions (54)

  • Theorem 2.0.1
  • proof
  • Corollary 2.0.2
  • Lemma 2.0.3
  • proof
  • Example 2.0.4
  • Example 2.0.5
  • Theorem 2.0.6
  • proof
  • Example 2.0.7
  • ...and 44 more