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

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 $k.$