Beraha's conjecture, wheels, and cyclic graphs

PolygonsA graph coloring assigns colors to the vertices of a graph in such a way that a pair of vertices joined by an edge do not get the same color. The chromatic polynomial of a graph gives the number of ways of coloring the graph with x colors. Beraha's numbers are B(n)=4*cos^2(pi/n). Tutte conjectured that there is a link between Beraha's numbers and some classes of graphs. This Demonstration shows that for a small number of vertices, it is not obvious what the connection is between the roots of the chromatic polynomial of a cyclic graph (green), the roots of the chromatic polynomial of the corresponding wheel graph (purple), and Beraha's numbers (red). However, taking more vertices clearly shows a relationship between these three sets of numbersComponente Curricular::Ensino Médio::Matemátic