Cayley sum graphs and eigenvalues of (3,6)(3,6)-fullerenes

We determine the spectra of cubic plane graphs whose faces have sizes 3 and 6. Such graphs, "(3,6)-fullerenes", have been studied by chemists who are interested in their energy spectra. In particular we prove a conjecture of Fowler, which asserts that all their eigenvalues come in pairs of the form $\{\lambda,-\lambda\}$ except for the four eigenvalues $\{3,-1,-1,-1\}$. We exhibit other families of graphs which are "spectrally nearly bipartite" in this sense. Our proof utilizes a geometric representation to recognize the algebraic structure of these graphs, which turn out to be examples of Cayley sum graphs

The connectivity of addition Cayley graphs

Abstract. For any finite abelian group G and any subset S ⊆ G, we determine the connectivity of the addition Cayley graph induced by S on G. Moreover, we show that if this graph is not complete, then it possesses a minimum vertex cut of a special, explicitly described form. 1. Background: addition Cayley graphs For a subset S of the abelian group G, we denote by Cay + G (S) the addition Cayley graph induced by S on G; recall that this is the undirected graph with the vertex set G and the edge set {(g1, g2) ∈ G × G: g1 + g2 ∈ S}. Note that S is not assumed to be symmetric, and that if S is finite, then Cay + G (S) is regular of degree |S | (if one considers each loop to contribute 1 to the degree of the corresponding vertex). The twins of the usual Cayley graphs, addition Cayley graphs (also called sum graphs) received much less attention in the literature; indeed, [A] (independence number), [CGW03] and [L] (hamiltonicity), [C92] (expander properties), and [Gr05] (clique number) is a nearly complete list of papers, known to us, where addition Cayley graphs are addressed. To some extent, this situation may be explained by the fact that addition Cayley graphs are rather difficult to study. For instance, it is well-known and easy to prove that any connected Cayley graph on a finite abelian group with at least three elements is hamiltonian, see [Mr83]; however, apart from the results of [CGW03], nothing seems to be known on hamiltonicity of addition Cayley graphs on finite abelian groups. Similarly, the connectivity of a Cayley graph on a finite abelian group is easy to determine, while determining the connectivity of an addition Cayley graph is a non-trivial problem, to the solution of which the present paper is devoted. The reader will see that investigating this problem leads to studying rather involved combinatorial properties of the underlying group