The chromatic polynomial of a graph counts the number of colorings of a graph as a function of the number of colors available. Here, a coloring intuitively means a way to color the vertices such that no two adjacent edges are the same color. For instance, for a triangle with k colors, the chromatic polynomial is k(k-1)(k-2) because there we can color the one vertex with k colors, leaving (k-1) choices for the second vertex, and accordingly (k-1) colors for the third vertex.
While studying algebraic graph theory with postdoc Mohamed Omar (now a professor at Harvey Mudd), I looked into existing research and wrote a report on the zeroes of chromatic polynomials. Interestingly, I found that chromatic polynomials cannot have negative zeroes, or zeroes in the intervals (0, 1) or (1, 32/27]. However, the zeroes are dense above 32/27, meaning that for any real number above 32/27, there is a chromatic polynomial for a graph with a root arbitrarily close to the chosen real number.
My report can be found here.