# Wheel graph and complete graph

• Sep 29th 2012, 06:56 PM
takaj
Wheel graph and complete graph
Hello, since the wheel graph with 3 vertices on the rim and 1 vertex in the middle is the same as the complete graph on 4 vertices, should I not be able to derive the general wheel graph chromatic polynomial for W3 from the general chromatic polynomial for a Complete graph(4)? .

I would write it up here, but it's pointless because all i am doing is expanding the elements inside of the brackets, but this is clearly not going to reach the general chromatic polynomial for a wheel graph.

Wheel graph Chromatic polynomial with 3 vertices on the rim is x(x-2)^3 +(-1)^3(x-2)

Complete graph on 4 vertices is x(x-1)(x-2)(x-3).

so should is not be the case that x(x-1)(x-2)(x-3)==x((x-2)^3 +(-1)^3(x-2)) ?

Edit: OK, i substituted x with a random number and then computed each polynomial and have gotten the same result from both polynomials with the same value for x, so they are clearly the same. I just need to show it algebraically.

Edit: OK, that was easy. I was clearly over thinking it. If i knew how to delete the thread i would.

Thanks.