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Mathematics

The Transfer Matrix Method for Counting Walks on Graphs

Quick fact

The transfer matrix method reduces walk-counting to matrix exponentiation: if M is the transfer matrix, then the (i,j) entry of M^k is exactly the number of walks of length k from vertex i to vertex j. This seemingly simple idea powers tools used in statistical mechanics and algorithm analysis.

Why this is interesting

You can often answer the question 'how many paths of length 10 are there?' with a simple matrix multiplication. But how can a matrix know about walks?