# Interlace Polynomials of Cycles with One Additional Chord

8-2018

Thesis

## Degree Name

Master of Science (MS)

## College/School

College of Science and Mathematics

## Department/Program

Mathematical Sciences

Aihua Li

Deepak Bal

Jonathan Cutler

## Abstract

In this research, we investigate the interlace polynomial of a certain type of cycle graph with additional edges, called chords. We focus on the graphs resulted by adding one chord to cycle graphs. Consider the cycle Cn with n edges. When adding one chord to it, two sub-cycles were created which share one edge. If the length of one sub-cycle is r (r ≥ 3), then the other length is n - r+2. All cycles with one chord resulting in a sub-cycle of length r, where r ≤ n - r + 2, are isomorphic, denoted by J(n,r). When n ≥ 4 and r = 3, we denote Mn = J(n, 3), for convenience. The main results of this thesis include iterative and explicit formulas for the interlace polynomial q(Mn, x) and properties of q(Mn, x) such as its degree, certain coefficients, and special values. An application in linear algebra derived from the adjacency matrix of Mn is explored. The interlace polynomial of J(n,r) is further investigated. Iterative formulas for q(J(n,r), x) are provided in the last chapter of the thesis.

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