Structural Analysis and Comparisons of Identity Graph of Groups

Abstract

The identity graphs of three groups are examined in this thesis: the quaternion group Q8, the symmetric group S5, and the alternating group A4. Every group element in an identity graph is a point (vertex) that is connected to both the identity and its inverse. We compare these graphs by examining fundamental characteristics such as the number of connections and points, triangles, self-inverse elements, and special groupings within the graph. Our research demonstrates how these graphs provide a clear and visual understanding of the group structure. This work proposes new techniques to use graphs to study groups by tying together concepts from group theory and graph theory. Due to vast usage of networks, numerous networks are considered and widely practiced in several branches of science, i.e in engineering, chemistry, biology, and computer networking. Networks can be describable in the aspect of graphs, where a vertex is affiliated with a node, and the connection between them as edges