Spectra of alternating-oriented tetragonal comb graphs

In this letter, we establish a recurrence relation for the characteristic polynomial of alternating-oriented tetragonal comb graphs. Using this relation, we express the polynomial in terms of Chebyshev polynomials of the second kind. This formulation allows us to explicitly determine the eigenvalues of the graphs. Furthermore, we analyze the geometric structure of the spectra in the complex plane and propose conjectures on generalized comb graphs involving higher-order polygons and various gluing configurations.