1

I have to study the resistance of the principal cluster of a percolating network of conducting wires . Individual wires are labeled from 1 to n. I represent the network by a graph G(V,E) and find its adjacency matrix A, where A_ij = 1 if wires i and j are in contact, 0 otherwise.

My question is the following : given that I need to implement Kirchhoff's Laws on the main percolated cluster, I need an algorithm that returns all the, ideally, smallest loops in the cluster. Do you know of an algorithm (mine is bruteforce now and not efficient) that finds all the loops inside a graph from its adjacency matrix ?

Improve this question

1 Answer 1

Reset to default
2

In general, there can be exponentially many simple cycles (loops), so since you want only the "smallest", it sounds as though you don't want them all. If you're looking to write equations corresponding to Kirchhoff's second law for all possible cycles, then it suffices to use just the equation for each cycle in a cycle basis. There is a polynomial-time algorithm to find the cycle basis that uses the least total number of edges (minimum cycle basis). Rather than implement that algorithm, however, it may suffice to switch from arc variables xu→v to differences of node variables yv - yu (fix one node variable per connected component to be zero).

Improve this answer
Sign up to request clarification or add additional context in comments.

1 Comment

Thanks for your answer. I really know close to nothing in graph theory. It's a bit 'shocked', as I guess these are concepts from linear algebra applied to graph theory, although I don't really see at first what constitutes the vector space. I guess I'll read up on the wiki article.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.