The graphs are used extensivelyin different applications for demonstrating the relations between the internalcomponents of the objects. Despite of the problems such as the lack of theefficient graph processing methods, there is still need to process graphs forrecognition of their structural pattern. The graph embedding in vector spacetries to insert the contexual and structural graph features into vectors andconsequently, make the rich and developed tools of statistical patternrecognition applicable to the graphs. Here, a grouping of the graph embeddingmethods is represented and among them, the spectral methods are investigated.The concept of graph signal processing is introduced which is emerged forbetter signal analysis through their internal relations. Then the operators andmulti-scale transforms of graph signal processing i vertex and spectrum scopes are introduced. Throughapplying these concepts, two graph embedding methods are proposed here, one forextracting the graph features from its different structural parts and anotherfor extracting its hierarchical structural features. In the first method,Generalized Frequency Filtering Embedding (GeFFE), applying the frequencyfiltering and the pseudo-Fourier operators is proposed, the first for using theeigenvalues and the second for using the eigenvector elements. The finalfeature vector is formed by selecting the best combinations of filter functionsand pseudo-Fourier operators in the underlying application. The experimentalresults show that GeFFE method enhances the classification accuracy in thetesed datasets, 7.55% in average, relative to the best performing previousmethod. Additionally, this proposed metoh resolves the co-spectrality problementirely in the tested datasets. In the second proposed method, DiffusionWavelet Embedding (DWE), a graph summarization method is proposed usingdiffusion wavelet as a multi-scale transform. In this summarization method, thenearest neighbor and maximum participant approaches are proposed for super-nodeconstruction and a vertex-identification based approach is proposed foradjacency matrix generation. The feature vector is made from every scaling andwavelet suaces by applying the basis functions on the initial graph andtaking the Laplacian spectrum from the resulting matrices. Through thisapproach, the eigenvalues and the eigenvectors are used at the same time forfeature vector extraction. Finally, the ensemble learning methods are appliedfor merging the classification results of the feature vectors of differentsuaces. The experiments confirms that using different levels of approximationand details in addition to the base level increases the classification accuracyand decreases the co-spectrality effect. DWE method enhances the classificationaccuracy in tested datasets, 6.25% in average, relative to the best-performingfprevious methods.