Resistors On A Dodecahedron

Each edge of a regular dodecahedron is a 1Ω1 \Omega resistor. If the effective resistance between two adjacent vertices can be represented as abΩ\dfrac {a}{b} \Omega where aa and bb are co-prime positive integers, find a+ba + b.

A dodecahedron has 20 vertices and 30 edges with 3 edges meeting at each vertex.


Image credit: Wikipedia.
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