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

TL;DR

This paper investigates the spectral properties of the generating graph of the cyclic group ng n, analyzing its adjacency and Laplacian spectra, and characterizes minimal generating sets of various sizes.

Contribution

It provides a detailed spectral analysis of the generating graph of ng n and explicitly characterizes all minimal generating sets of given sizes.

Findings

Spectral bounds for the adjacency matrix of ng n's generating graph.

Spectral bounds for the Laplacian matrix of the generating graph.

Complete characterization of minimal generating sets of ng n of size k.

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 $\Gamma (G)$ 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 $\Gamma(\mathbb{Z}_n)$ 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 $k.$