Sam Northshield

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We give three proofs that the reciprocal of Ihara's zeta function can be expressed as a simple polynomial times a determinant involving the adjacency matrix of the graph. The rst proof, for regular graphs, is based on representing radial symmetric eigenfunctions on regular trees in terms of certain polynomials. The second proof, also for regular graphs, is(More)
Imagine a sphere with its equator inscribed in an equilateral triangle. This Saturn-like figure will help us understand from where Cardano's formula for finding the roots of a cubic polynomial p(z) comes. It will also help us find a new proof of Marden's theorem, the surprising result that the roots of the derivative p (z) are the foci of the ellipse(More)
Newton's method applied to a quadratic polynomial converges rapidly to a root for almost all starting points and almost all coefficients. This can be understood in terms of an associative binary operation arising from 2 × 2 matrices. Here we develop an analogous theory based on 3 × 3 matrices which yields a two-variable generally convergent algorithm for(More)
1. INTRODUCTION. This paper has its genesis in a problem the author first came upon while in college. Although the areas covered here are well travelled and nothing here is guaranteed original, it covers a pleasant nexus of many mathematical strands. Furthermore, we show the value of good notation and of reading an old master for solving a problem. Consider(More)
ABSRACT. For a graph G, the " growth " of G is the constant c such that metric balls in the universal covering tree T grow like c n. The " cogrowth " of G is the constant c such that the intersection of the preimage in T of a vertex of G with metric balls grow with order c n. For example, the growth of a d-regular graph is d − 1 and the cogrowth of G is(More)
We introduce a sequence b(n) of algebraic integers that is an analogue of Stern's diatomic sequence, not only in definition, but also in many of its properties. Just as Stern's sequence arises from Ford circles, so too b(n) arises from an array of circles. We study the generating function for b(n) and derive several closed formulas for the sequence. Two(More)