The spectrum of the vertex quadrangulation of a 4-regular toroidal graph and beyond

Let G be a simple plane graph with all vertices of valency 4. The vertex quadrangulation QG of G visually looks like a graph whose vertices are depicted as empty squares, and the connecting edges are attached to the corners of the squares. In particular, such is the molecular graph of the putative polymer of cyclobutadiene. This graph can be represented in finite form as a snippet of wallpaper or as a graph embedded in the surface of a torus. We consider the spectra of eigenvalues of toroidal vertex-quadrangulated graphs, also casting a glance at some related issues. Finding such spectra is possible due to using known spectral methods specially designed for symmetric graphs. However, the analytical calculation of the spectrum of an arbitrary vertex-quadrangulated graph (not necessarily with symmetry) still remains an unsolved problem. The current work is also a preliminary exploratory consideration of this complex problem.