## Fowler's Conjecture on eigenvalues of (3,6)-polyhedra

A (*k*,6)-polyhedron is a cubic graph embedded in the plane so that all
of its faces are *k*-gons or hexagons. Such graphs exist only for *k*
= 2,3,4,5. The (5,6)-polyhedra are also known as fullerene graphs since they
correspond to the molecular graphs of fullerenes.

The (3,6)-polyhedra have precisely 4 triangular faces and they cover the
complete graph K_{4}. Therefore, the eigenvalues 3, -1, -1, -1 of K_{4}
are also eigenvalues of every (3,6)-polyhedron. Patrick Fowler computed
eigenvalues of numerous examples and observed that all other eigenvalues occur
in pairs of opposite values *x*, -*x*, a similar phenomenon as for
bipartite graphs. From the spectral information, the (3,6)-polyhedra therefore
behave like a combination of K_{4} and a bipartite graph.

**Fowler's Conjecture:** Let
G be the graph of a (3,6)-polyhedron with 2*k *+ 4 vertices. Then the
eigenvalues of G can be partitioned into threee classes: K = {3, -1, -1, -1}, P
= {*x*_{1}, ..., *x*_{k}} (where *x*_{i}
is nonnegative for *i* = 1, ..., *k*), and N = - P.

Horst Sachs and Peter John (private communication) found some reduction
procedures which allow Fowler's Conjecture to be proved for many infinite
classes of (3,6)-polyhedra.

Added (april 2002): See also

[1] P. W. Fowler, P. E. John, H. Sachs,
(3,6)-cages, hexagonal toroidal cages, and their spectra,
Discrete mathematical chemistry (New Brunswick, NJ, 1998), pp. 139-174,
DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 51,
Amer. Math. Soc., Providence, RI, 2000.

Send comments to Bojan.Mohar@uni-lj.si

##### Revised: april 08, 2002.