## 1 Introduction  
Recent reports from the World Health Organization (WHO), including the Intersectoral Global Action Plan (IGAP) on epilepsy and neurological disorders [1], emphasise advancing methodologies to improve diagnosis, treatment, and care for patients with cognitive and neurological disorders. By capturing spatiotemporal brain dynamics, Electroencephalography (EEG) enables the study of functional connectivity patterns and metastable states for investigating resting-state brain functions and event-related neural dynamics [2]. However, the analysis of EEG signals presents significant challenges due to their inherent nonlinearity, non-stationarity, and low signal-to-noise ratio (SNR). These challenges are exacerbated by EEG data sparsity and high dimensionality, necessitating advanced analytical techniques to extract meaningful insights. Class imbalance in EEG datasets is another challenge, where critical events such as seizures or high cognitive loads are underrepresented. The analysis of latent metastable substructures in EEG connectivity remains a critical gap in the field. Metastructures, such as metastable connectivity states, represent transient but meaningful neural dynamics that are key to understanding cognitive processes [3]. Deep learning methods have exploited linear embeddings of nonlinear dynamics [4]. However, current methods for EEG analysis often fail to fully exploit the latent metastable structures within brain networks, particularly in highly nonlinear, non-stationary datasets, and often fail to account for this imbalance, leading to biased models that favour majority classes, compromising clinical applications [5; 6]. Spectral analysis of the Koopman operator, combined with graph kernels, has emerged as a promising framework for uncovering these latent patterns. Koopman operators transform nonlinear dynamical systems into linear ones, enabling interpretable embeddings, while graph kernels encode connectivity networks' structural and temporal properties [7; 8; 9]. Recent research has exploited the potential of combining advanced graph theoretical methods [10; 11; 12] and dynamical system modelling [13; 14]. Despite advancements in graph-based feature extraction, systematic investigations of the efficacy and robustness of graph kernels for EEG data analysis are limited [15]. Graph kernels, such as the Weisfeiler-Lehman (WL) kernel, Random Walk (RW) kernel, and Graphlet kernel, offer computational efficiency and structural expressiveness but remain underexplored in their ability to extract latent metastable patterns. Moreover, their integration with Koopman spectral operators for EEG analysis represents an untapped opportunity for advancing the field [5; 16]. Koopman operators provide linear embeddings of nonlinear dynamical systems, and when approximated through graph kernels, they enable a robust spectral decomposition of connectivity structures. However, a critical gap exists in the literature regarding the systematic evaluation of graph kernels for their robustness and efficacy in estimating such metastructures of connectivity. This study proposes a novel workflow that integrates graph kernel-based Koopman spectral operators for feature extraction and classification, specifically addressing these gaps in the context of EEG datasets. **Existing approaches for EEG analysis**--Numerous methods have been proposed for feature extraction and classification of EEG signals. Still, significant limitations persist in their ability to capture and explain complex connectivity dynamics. **Feature extraction techniques**--[PERSON] et al. (2021) used Dynamic Mode Decomposition (DMKD) and Empirical Mode Decomposition (EMD) for feature extraction in seizure detection, achieving high sensitivity and specificity using a Multilayer Long Short-Term Memory (LSTM) Discriminant Neural Network. However, the study lacked the interpretability of discriminant features and relied on basic LSTM models without further architectural exploration [6]. [PERSON] et al.