Permutability graphs of subgroups of some finite non-abelian groups

TL;DR

This paper explores the structure and properties of permutability graphs of subgroups in specific finite non-abelian groups, analyzing various graph invariants and characteristics.

Contribution

It provides a detailed study of permutability graphs for dihedral, quaternion, quasi-dihedral, and modular groups, including their structural and combinatorial properties.

Findings

Determined the number of edges and degrees of vertices in the graphs.

Analyzed graph invariants such as independence, domination, and clique numbers.

Investigated properties like Eulerian and Hamiltonian cycles.

Abstract

In this paper, we study the structure of the permutability graphs of subgroups, and the permutability graphs of non-normal subgroups of the following groups: the dihedral groups $D_n$, the generalized quaternion groups $Q_n$, the quasi-dihedral groups $QD_{2^n}$ and the modular groups $M_{p^n}$. Further, we investigate the number of edges, degrees of the vertices, independence number, dominating number, clique number, chromatic number, weakly perfectness, Eulerianness, Hamiltonicity of these graphs.