This paper studies the \(k^{\text {th}}-\) order slant Toeplitz and slant little Hankel operators on the weighted Bergman space \(\mathcal {A}_\alpha ^2(\mathbb {D})\) . We derive their matrix representations with respect to the standard orthonormal basis and establish criteria for boundedness and compactness. For slant Toeplitz operators, we prove that compactness occurs if and only if the symbol is identically zero. For slant little Hankel operators, we provide a necessary and sufficient condition for compactness in terms of the symbol’s coefficients. We further demonstrate that these operators generally fail to be normal and do not commute. In addition, we investigate their spectral properties, including a detailed description of their essential spectra and eigenvalue distribution. Finally, we introduce a novel graph-theoretic framework that visualizes the structural and combinatorial properties of these operators, offering a new perspective on their operator-theoretic behavior.
