Sheffer-type lacunary bivariate $ q $-Laguerre polynomials with applications to heterogeneous graph filters

We introduce a lacunary bivariate $ q $-Laguerre–Sheffer family generated by the product of a Sheffer factor, a first $ q $-exponential, and a $ q $-Tricomi–Bessel kernel. The construction has three independent controls: the lacunarity index fixes which powers enter the finite sums, the kernel order changes the Bessel-type shifts, and the Sheffer factor supplies an additional deformation of the sequence. We derive explicit coefficient and convolution formulas, lowering relations, a quasi-monomial operator structure, a $ q $-difference equation, and a $ q $-translation identity. We then evaluate the polynomials at a mass-weighted graph Laplacian associated with heterogeneous network diffusion. This gives polynomial spectral filters whose vertex-domain support is limited to a prescribed graph-hop radius. Numerical examples illustrate the locality bound, spectral response tuning, comparison with a Chebyshev polynomial filter, kernel truncation accuracy, and a one-parameter calibration problem.