Elementary excitations of topologically disordered systems

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We have examined the elementary excitation spectra of continuous random networks having only even rings or both even and odd rings. We have shown that odd rings introduce localized states and suppress the density of states near a spectral bound associated with sign alteration in the spatial dependence of the state. While this is a general conclusion, we show it explicitly only for the nearest-neighbor tight-binding s-band and related magnon and phonon models. In the latter case, phonon softening or phonon hardening can result. In the former case, we show that frustration leads to localized magnons.