Date of Award

2014

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Woong Kook

Abstract

Given a finite connected planar graph G with s finite faces, we define the cycle-intersection matrix, C(G) = (cij) to be a symmetric matrix of size s [1] s where cii is the length of the cycle which bounds finite face i, and cij is the negative of the number of common edges in the cycles bounding faces i and j for i 6= j. We will show that detC(G) equals the number of spanning trees in G. As an application, we compute the number of spanning trees of grid graphs via Chebychev polynomials. In addition, we show an interesting connection between the determinant of C(G) to the Fibonacci sequence when G is a triangulation of an n-gon by non-overlapping diagonals. We also apply methods from graph theory to the field of post-secondary mathematics education. We describe here a remediation program designed to help calculus students fill in the gaps in their precalculus knowledge. This program has provided us with a way to strengthen the quantitative skills of our students without requiring a separate course. The data collected are analyzed here and suggestions for program improvement are made.

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