Human brain functional connectivity (FC) networks, estimated based on resting-state functional magnetic resonance imaging (rs-fMRI), has become a promising tool for imaging-based brain disease diagnosis. Conventional low-order FC network (LON) usually characterizes pairwise temporal correlation of rs-fMRI signals between any pair of brain regions. Meanwhile, high-order FC network (HON) has provided an alternative brain network modeling strategy, characterizing more complex interactions among low-order FC sub-networks that involve multiple brain regions. However, both LON and HON are usually constructed within a fixed and relatively wide frequency band, which may fail in capturing (sensitive) frequency-specific FC changes caused by pathological attacks. To address this issue, we propose a novel "multi-frequency HON construction" method. Specifically, we construct not only multiple frequency-specific HONs (intra-spectrum HONs), but also a series of cross-frequency interaction-based HONs (inter-spectrum HONs) based on the low-order FC sub-networks constructed at different frequency bands. Both types of these HONs, together with the frequency-specific LONs, are used for the complex network analysis-based feature extraction, followed by sparse regression-based feature selection and the classification between mild cognitive impairment (MCI) patients and normal aging subjects using a support vector machine. Compared with the previous methods, our proposed method achieves the best diagnosis accuracy in early diagnosis of Alzheimer's disease.