# Jazmin Ortiz

## Thesis

Chromatic Polynomials, Orbital Chromatic Polynomials, and Their Roots

- Advisor
- Mohamed Omar
- Second Reader(s)
- Michael Orrison

## Abstract

The chromatic polynomial, $P_\Gamma(x)$ of a graph $\Gamma$, is a polynomial that when evaluated at a positive integer $k$, is the number of proper $k$ colorings of the graph $\Gamma$. We can then find the orbital chromatic polynomial $OP_{\Gamma, G}(x)$ of a graph $\Gamma$ and a group $G$ of automorphisms of $\Gamma$, which is a polynomial whose value at a positive integer $k$ is the number of orbits of $k$-colorings of a graph $\Gamma$ when acted upon by the group $G$. By considering the roots of the orbital chromatic and chromatic polynomials, the similarities and differences of these polynomials is studied. Specifically we work toward proving a conjecture concerning the gap between the real roots of the chromatic polynomial and the real roots of the orbital chromatic polynomial.