On the metric dimension of the character degree graph of a solvable group

TL;DR

This paper determines the metric dimension of various specific character degree graphs of finite solvable groups, providing insights into their structural properties and related graph parameters.

Contribution

It calculates the metric dimension for several classes of character degree graphs of solvable groups, including regular, diameter 2 and 3, and Fitting height 2 graphs, and explores related parameters.

Findings

Metric dimension for regular character degree graphs.

Metric dimension for diameter 2 non-block graphs.

Metric dimension for diameter 3 graphs with a cut vertex.

Abstract

Let $G$ be a finite solvable group and let $\Delta(G)$ be the character degree graph of $G$. In this paper, we obtain the metric dimension of certain character degree graphs. Specifically, we calculate the metric dimension for a regular character degree graph, a character degree graph with a diameter of $2$ that is not a block, a character degree graph with a diameter of $3$ that also has a cut vertex and a character degree graph with Fitting height $2.$ We also consider two related parameters, base size and adjacency dimension, and their relation to metric dimension for character degree graphs of solvable groups.