Joel Friedman

In this thesis, we discuss modelling with (virtual) k-diagrams a class of functions from ℤⁿ to ℤ, which we call Riemann functions, that generalize the graph Riemann-Roch rank functions of Baker and Norine. The graph Riemann-Roch theorem has seen significant activity in recent years, however, as of yet, there is not a satisfactory homological interpretation of this theorem. This may be viewed as disappointing given the rich homological theory contained in the classical Riemann-Roch theorem. For a graph Riemann-Roch function, we will be able to express the associated graph Riemann-Roch formula as a (virtual) Euler characteristic of the modelling (virtual) k-diagram. From the development of this approach, we obtain a new formula for computing the graph Riemann-Roch rank of the complete graph Kn.

View record

The girth of a graph is the length of the shortest cycle in a graph, and the abelian girth of a graph is the girth of the graph's universal abelian covering graph. We denote the abelian girth of a graph G as Abl(G) and show that for d-regular graphs on n vertices with d constant and n growing we have Abl(G) ≤ 6 log_{d-1} n plus a vanishing term. This can be seen as a version of the Moore bound for abelian girth. We also prove Girth(G) ≤ Abl(G)/3, which implies that any multiplicative improvement to the abelian girth Moore bound would also improve the standard Moore bound. Sheaves on graphs and two of their homological invariants, the maximum excess and the first twisted Betti number, were used in the proof of the Hanna Neumann Conjecture from algebra and may be of use in proving several related unresolved conjectures. These conjectures can be proven if certain sheaves called ρ-kernels have vanishing maximum excess. Ungapped sheaves have maximum excess equal to the first twisted Betti number, and it is easy to compute the maximum excess of a given sheaf in the case that the sheaf is not gapped. For general sheaves though, there is no known way of computing the maximum excess in polynomial time. We give several conditions that gapped sheaves must satisfy. These conditions include that a gapped sheaf must have edge dimension at least as large as the abelian girth of the underlying graph. The ρ-kernels are subsheaves of constant sheaves. We prove that gapped subsheaves of constant sheaves exist, implying that finding maximum excess of some ρ-kernels may be computationally difficult.

View record

For a fixed graph, B, we study a probability model of random covering maps of degree n. Specifically, we study spectral properties of new eigenvalues of the adjacency matrix of a random covering, and its Hashimoto matrix (i.e., the adjacency matrix of the associated directed line graph).Our main theorem says that if B is d-regular, then for every positive epsilon, the probability that a random covering has a new adjacency eigenvalue greater than 2(d-1)^(1/2) + epsilon tends to zero as n tends to infinity. This matches the generalized Alon-Boppana lower bound.For general base graphs, B, Friedman conjectured in that the new eigenvalue bound holds with 2(d-1)^(1/2) replaced with the spectral radius of the universal cover of B. We refer to this conjecture as the generalized Alon conjecture; Alon stated this conjecture in the case where B has one vertex, i.e., the case of random d-regular graphs on n vertices. However, for some non-regular base graphs B, we cannot yet prove any non-trivial new eigenvalue upper bound with high probability.We use trace methods, as pioneered by Broder and Shamir for random, d-regular graphs; these methods were eventually refined by Friedman to prove the original Alon conjecture, i.e., in the case where B has one vertex. Our methods involve several significant simplifications of the methods of Friedman.

View record