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Tutte polynomial of a cycle
Task number: 4067
Determine the Tutte polynomial for the cycle of length k.
Hint
Use the recurrence for the Tutte polynomial TG(x,y).
TG={x⋅TG−ewhen e is a bridge,y⋅TG−ewhen e is a loop, TG−e+TG/ewhen e is not a bridge nor a loop.Furthermore, an edgeless graph has TG=1.
Solution
In the first step, the last option is used. So TCk=TPk−1+TCk−1.
By removing leaves, we find that TPk−1=xk−1, which leads to the recurrence TCk=xk−1+TCk−1
Lastly, we apply recurrence to the bigon, i.e. to two vertices connected by two edges. Then one vertex is left with a loop C1, where TC1=y.
Answer
The Tutte polynomial of a cycle is: TCk(x,y)=xk−1+xk−2+⋯+x2+x+y.