This work explores new graphs constructed on generalized Quaternion groups  The vertices are the conjugacy classes of  and there will be an edge between the two conjugacy classes if their orders are distinct. We define this simple graph as order structured conjugacy class simple graph (OSCSG), labelled by  In this research, we characterize the proposed graph and prove that for  the closure of  can never be a complete graph. Its complement,  always contains a maximum complete subgraph  for  Moreover, for  the upper bound of average distance is determined. Finally, the bipartite and tri-partite nature of these graphs along with their spectral characteristics is also discussed.

structured graphs, vertex subsets with special properties, order based graphs.

Received: November 29, 2024; Revised: February 17, 2025; Accepted: March 3, 2025; Published: May 24, 2025

How to cite this article: Aneela, M. Khalid Mahmood and Daud Ahmad, A special class of graphs emerging from quaternion groups, JP Journal of Algebra, Number Theory and Applications 64(4) (2025), 379-394. https://doi.org/10.17654/0972555525020

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