hosoya polynomial of cayley graphs of d2n and q2n with a focus on specific generating sets

Cayley graphs are powerful tools for analyzing algebraic groups through their combinatorial and topological properties. in this paper, we investigate the hosoya polynomial of cayley graphs associated with the dihedral group d2n and the generalized quaternion group q2n , considering specific generating subsets. general formulas for computing these polynomials are presented, and their connection with the wiener index is examined. our results show that both the hosoya polynomial coefficients and the wiener index exhibit exponential growth with group order, reflecting the increasing complexity of the graphs. this study contributes to the understanding of distance-based invariants in algebraic graph theory and provides a foundation for further exploration in network analysis