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A SPECIAL CLASS OF GRAPHS EMERGING FROM QUATERNION GROUPS
Keywords:
structured graphs, vertex subsets with special properties, order based graphsDOI:
https://doi.org/10.17654/0972555525020Abstract
This work explores new graphs constructed on generalized Quaternion groups $Q_{4 n}$. The vertices are the conjugacy classes of $Q_{4 n}$ 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 $\Gamma_{c l}^o\left(Q_{4 n}\right)$. In this research, we characterize the proposed graph and prove that for $n>5$, the closure of $\Gamma_{c l}^o\left(Q_{4 n}\right)$ can never be a complete graph. Its complement, $\overline{\Gamma_{c l}^o\left(Q_{4 n}\right)}$ always contains a maximum complete subgraph $K_{n-1}$ for $n>2$. Moreover, for $n>5$, 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.
Received: November 29, 2024
Revised: February 17, 2025
Accepted: March 3, 2025
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