Non-solvable graphs of groups

TL;DR

This paper explores the properties of the non-solvable graph associated with a group, analyzing its structure, connectivity, and realizability, and compares graphs of different groups to understand their algebraic implications.

Contribution

It provides a comprehensive study of the non-solvable graph's properties and introduces new results on its structure, connectivity, and realizability, including the impact of group isomorphisms.

Findings

Determined vertex degree and its cardinality in the non-solvable graph.

Established the non-planarity and non-toroidality of the non-solvable graph.

Analyzed properties of groups with isomorphic non-solvable graphs.

Abstract

Let $G$ be a group and $Sol(G)=\{x \in G : \langle x,y \rangle \text{ is solvable for all } y \in G\}$. We associate a graph $\mathcal{NS}_G$ (called the non-solvable graph of $G$) with $G$ whose vertex set is $G \setminus Sol(G)$ and two distinct vertices are adjacent if they generate a non-solvable subgroup. In this paper we study many properties of $\mathcal{NS}_G$. In particular, we obtain results on vertex degree, cardinality of vertex degree set, graph realization, domination number, vertex connectivity, independence number and clique number of $\mathcal{NS}_G$. We also consider two groups $G$ and $H$ having isomorphic non-solvable graphs and derive some properties of $G$ and $H$. Finally, we conclude this paper by showing that $\mathcal{NS}_G$ is neither planar, toroidal, double-toroidal, triple-toroidal nor projective.