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We show that logarithmic derivative of the Zeta function of any regular graph is given by a power series about infinity whose coefficients are given in terms of the traces of powers of the graph's Hashimoto matrix. We then consider the expected value of this power series over random, $d$-regular graph on $n$ vertices, with $d$ fixed and $n$ tending to infinity. Under rather speculative assumptions, we make a formal calculation that suggests that for fixed $d$ and $n$ large, this expected value should have simple poles of residue $-1/2$ at $\pm (d-1)^{-1/2}$. We shall explain that calculation suggests that for fixed $d$ there is an $f(d)>1/2$ such that a $d$-regular graph on $n$ vertices is Ramanujan with probability at least $f(d)$ for $n$ sufficiently large. Our formal computation has a natural analogue when we consider random covering graphs of degree $n$ over a fixed, regular "base graph." This again suggests that for $n$ large, a strict majority of random covering graphs are relatively Ramanujan. We do not regard our formal calculations as providing overwhelming evidence regarding the frequency of Ramanujan graphs. However, these calculations are quite simple, and yield intiguing suggestions which we feel merit further study.

We study the rigidity of body-and-cad frameworks which capture the majority of the geometric constraints used in 3D mechanical engineering CAD software. We present a combinatorial characterization of the generic minimal rigidity of a subset of body-and-cad frameworks in which we treat 20 of the 21 body-and-cad constraints, omitting only point-point coincidences. While the handful of classical combinatorial characterizations of rigidity focus on distance constraints between points, this is the first result simultaneously addressing coincidence, angular, and distance constraints. Our result is stated in terms of the partitioning of a graph into edge-disjoint spanning trees. This combinatorial approach provides the theoretical basis for the development of deterministic algorithms (that will not depend on numerical methods) for analyzing the rigidity of body-and-cad frameworks., Comment: 17 pages, 7 figures, version to appear in Symposium on Solid and Physical Modeling '12 and associated special issue of Computer Aided Design