We introduce a new invariant, the coronal of a graph, and use it to compute the spectrum of the corona G • H of two graphs G and H. In particular, we show that this spectrum is completely determined by the spectra of G and H and the coronal of H. Previous work has computed the spectrum of a corona only in the case that H is regular. We then explicitly compute the coronals for several families of graphs, including regular graphs, complete n-partite graphs, and paths. Finally, we use the corona construction to generate many infinite families of pairs of cospectral graphs.

Extending the work of Godsil and others, we investigate the notion of the inverse of a graph (specifically, of bipartite graphs with a unique perfect matching). We provide a concise necessary and sufficient condition for the invertibility of such graphs and generalize the notion of invertibility to multigraphs. We examine the question of whether there exists a "litmus subgraph" whose bipartiteness determines invertibility. As an application of our invertibility criteria, we quickly describe all invertible unicyclic graphs. Finally, we describe a general combinatorial procedure for iteratively constructing invertible graphs, giving rise to large new families of such graphs.

Abstract. The number F (h) of imaginary quadratic fields with class number h is of classical interest: Gauss' class number problem asks for a determination of those fields counted by F (h). The unconditional computation of F (h) for h ≤ 100 was completed by M. Watkins, using ideas of Goldfeld and Gross-Zagier; Soundararajan has more recently made conjectures about the order of magnitude of F (h) as h → ∞ and determined its average order. In the present paper, we refine Soundararajan's conjecture to a conjectural asymptotic formula and also consider the subtler problem of determining the number F (G) of imaginary quadratic fields with class group isomorphic to a given finite abelian group G. Using Watkins' tables, one can show that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance (Z/3Z) 3 does not). This observation is explained in part by the Cohen-Lenstra heuristics, which have often been used to study the distribution of the p-part of an imaginary quadratic class group. We combine heuristics of Cohen-Lenstra together with our refinement of Soundararajan's conjecture to make precise predictions about the asymptotic nature of the entire imaginary quadratic class group, in particular addressing the above-mentioned phenomenon of "missing" class groups, for the case of p-groups as p tends to infinity. Furthermore, conditionally on the Generalized Riemann Hypothesis, we extend Watkins' data, tabulating F (h) for odd h ≤ 10 6 and F (G) for G a p-group of odd order with |G| ≤ 10 6 . The numerical evidence matches quite well with our conjectures.

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A classical result of Dirichlet shows that certain elementary character sums compute class numbers of quadratic imaginary number fields. We obtain analogous relations between class numbers and a weighted character sum associated to a 2-isogeny of elliptic curves.We consider the following point of view for Dirichlet's result. The F p -rational mor-p into two sets, those that are the image of an F p -rational point of the domain (that is, the quadratic residues) and those that are not (the non-residues). The character χ φ := (·/p) is now precisely the natural identification of the cokernel of φ with {±1} which makes the following sequence exact:Note that it is the properties of φ, not the underlying algebraic group G m , which allow this construction. This paper demonstrates that an analogous procedure, arising from a different morphism of algebraic groups, yields new character sums with similar arithmetic properties.