CYCLIC INTERSECTION GRAPH OF SUBGROUPS AND p-SUBGROUPS GRAPH OF SOME FINITE GROUPS WITH THEIR PROPERTIES

Abstract

Various techniques have been used by researchers to study the properties of groups. One such technique is the graph theoretical approach. This technique creates a bridge from group theory to graph theory, allowing an investigation of the properties of groups in terms of graphs. In this research, a cyclic intersection graph of the subgroups of a finite group is newly defined as a graph that has all the non-trivial subgroups of the finite groups as its vertices and any two vertices are adjacent if and only if their intersection is a non-trivial cyclic subgroup. Another graph called the 𝑝-subgroup graph of a group is also newly defined as a graph whose vertices are the elements of a group and whose edges connect pairs of vertices that generate a 𝑝-subgroup. The cyclic intersection graphs of the subgroups are determined for the dihedral groups, the generalized quaternion groups, and the quasi-dihedral groups using the definition. The graphs are found to be connected. Meanwhile, the 𝑝-subgroup graph is also found for those three types of groups by using the newly defined graph. The 𝑝-subgroup graph is connected for all values of 𝑛 ≥ 4 in quasi-dihedral groups. However, it is connected for dihedral groups and generalized quaternion groups only when the degree is a power of two. The 𝑝-subgroup graph is not planar for any quasi-dihedral group, but it is planar for dihedral groups with degree 3 or degree 6, and for generalized quaternion groups with order 12. The properties under examination play a pivotal role in deriving several graph invariants associated with dihedral groups, generalized quaternion groups, and quasi- dihedral groups. These invariants encompass characteristics such as the clique number, independent number, domination number, girth, diameter, vertex chromatic number, and edge chromatic number. Moreover, this study delves into various topological indices, including the Wiener index and the first and second Zagreb index, concerning the cyclic intersection graphs of subgroups and 𝑝-subgroup graphs within specific finite groups. Specifically, the study presents the topological indices for the cyclic intersection graph of subgroups in cases where 𝑛 takes the form of 𝑝𝑟 , 𝑝𝑟 𝑞, and 𝑝𝑞ℎ, where 𝑝, 𝑞, and ℎ are distinct primes. Additionally, the research computes the topological indices for the 𝑝-subgroup graphs of dihedral groups of degrees 2𝑟 and 𝑝𝑟 , generalized quaternion groups of degree 2𝑟 , and quasi-dihedral groups of all values of 𝑛 ≥ 4.