On the generator graph of a cyclic group

TL;DR

This paper investigates the properties of the generator graph of cyclic groups, establishing bounds on its diameter, formulas for topological indices, and calculating its metric dimension.

Contribution

It extends the study of generator graphs by providing new bounds, explicit formulas, and metric dimension calculations specifically for cyclic groups.

Findings

Diameter of generator graph is at most 2.

Explicit formulas for topological indices are derived.

Metric dimension of the generator graph is determined.

Abstract

In this paper, we continue the study of the generator graph of a group. In 2023, Tacbobo [9] defined the generator graph of a nontrivial group to be the graph whose vertices are the elements of the group, with two vertices being adjacent if at least one of them is a generator of the group. Building on the properties established in [9], we prove that the diameter of the generator graph of a cyclic group is at most $2$. Furthermore, we present explicit formulas for some topological indices of the generator graph of a cyclic group with $n \ge 2$ elements and whose set of generators is $S$, expressed in terms of $n$ and $|S|$. Lastly, we determine the metric dimension of the generator graph of a nontrivial cyclic group as a function of its order $n$.