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We study AKLT models on locally tree-like lattices of fixed connectivity andfind that they exhibit a variety of ground states depending upon the spin,coordination and global (graph) topology. We find a) quantum paramagnetic orvalence bond solid ground states, b) critical and ordered N\'eel states onbipartite infinite Cayley trees and c) critical and ordered quantum vector spinglass states on random graphs of fixed connectivity. We argue, in consonancewith a previous analysis, that all phases are characterized by gaps to localexcitations. The spin glass states we report arise from random long rangedloops which frustrate N\'eel ordering despite the lack of randomness in thecoupling strengths.

AKLT Models with Quantum Spin Glass Ground States