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The geometrically frustrated spin-1/2 Ising-Heisenberg model on triangulatedHusimi lattices is exactly solved by combining the generalized star-triangletransformation with the method of exact recursion relations. The ground-stateand finite-temperature phase diagrams are rigorously calculated along with bothsublattice magnetizations of the Ising and Heisenberg spins. It is evidencedthat the Ising-Heisenberg model on triangulated Husimi lattices with two orthree inter-connected triangles-in-triangles units displays in a highlyfrustrated region a quantum disorder irrespective of temperature, whereas thesame model on triangulated Husimi lattices with a greater connectivity oftriangles-in-triangles units exhibits at low enough temperatures an outstandingquantum order due to the order-by-disorder mechanism. The quantum reduction ofboth sublattice magnetizations in the peculiar quantum ordered state graduallydiminishes with increasing the coordination number of underlying Husimilattice.

Is order-by-disorder present or absent in a highly frustrated region of the spin-1/2 Ising-Heisenberg model on triangulated Husimi lattices?