Resistors On A Dodecahedron

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

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


Image credit: Wikipedia.
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