#### Date of Award

2014

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics

#### 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)* = (c* _{ij}*) to be a symmetric matrix of size

*s x s*where

*c*is the length of the cycle which bounds finite face

_{ii}*i*, and

*c*is the negative of the number of common edges in the cycles bounding faces

_{ij}*i*and

*j*for

*i*≠

*j*. We will show that det

*C(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.

#### Recommended Citation

Phifer, Caitlin R., "THE CYCLE INTERSECTION MATRIX AND APPLICATIONS TO PLANAR GRAPHS AND DATA ANALYSIS FOR POSTSECONDARY MATHEMATICS EDUCATION" (2014). *Open Access Dissertations.* Paper 210.

https://digitalcommons.uri.edu/oa_diss/210

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