Tutte polynomial of a multiedge
Task number: 4068
Determine the Tutte polynomial for two vertices connected by \( k \) edges.
Hint
Use the recurrent relation to calculate the Tutte polynomial.
Solution
Let us denote the two vertices connected by \( k \) edges by \( B_k \). Next, let’s denote the vertex with \( k \) loops as \( L_k \).
In the first step, the last option is used, i.e. \( T_{B_k} = T_{B_{k-1}} + T_{L_{k-1}} \).
By removing the loops, we find that \( T_{L_{k-1}} = y^{k-1} \).
The recurrence stops after we apply it on the bigon (two vertices connected by two edges). Then we obtain a path \( P_1 \). We apply the first rule where \( T_{P_1} = x \).
Answer
Tutte polynomial of a multi-edge is \( T_{B_k} (x, y) = x + y^{k-1} + y^{k-2} + \cdots + y^2 + y \).
