A (p,q)-cage is (the graph of) a simple (i.e., trivalent) polyhedron each face of which is a p-gon or a q-gon. (5,6)-cages correspond to the spherical fullerenes; (3,6)-cages are closely related to the toroidal fullerenes (trivalent hexagonal tessellations of the torus), their structure can easily and intuitively be described by only three parameters (see [1]).

In the talk, the spectrum of a (3,6)-cage, including a central open problem (P.W. Fowler's conjecture, see [1], p. 159), will be discussed.

[1] P.W. Fowler, P.E. John and H.Sachs,(3,6)-cages, hexagonal toroidal cages, and their spectra. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 51 (2000), 139-174.