A combinatorial proof of the Rayleigh formula for graphs

A combinatorial proof of Rayleigh monotonicity for graphs

“…The corresponding entries of the third and fourth columns describe the pair (A , B ) associated with (A, B) that contributes to equation (3) for H. It is easy to see that the pairs (A, B) and (A , B ) have the same type. Thus, the construction described in Table 1 gives a 1:1 correspondence (A, B) ↔ (A , B ) that proves equations ( 4) and (5). Therefore, we have verified equation (3) for G in Case 3.…”

“…By the induction hypothesis, equation (3) holds for each of the graphs H(ab), H(ac), and H(bc), so that (4) and (5) suffice to prove equation (3) for G.…”

“…The case of general conductances y > 0 on graphs can be found in Section 3.8 of Balabanian and Bickart [1]. Choe [4,5,6] gives two proofs of Theorem 1. One -based on Tutte's theory of chain groups as in [10] -generalizes to all sixth-root-of-unity matroids but gives a less precise description of the right-hand side.…”

Monotonic and Non-Monotonic Epidemiological Models on Networks

“…This theorem is an epidemiological equivalent of a much-studied property of electrical networks, known as Rayleigh monotonicity [13][14][15]. Rayleigh monotonicity describes the effect of individual resistors in a network on the "effective resistance" of the entire network.…”

Elementary proof of Rayleigh formula for graphs

“…Take any partition (F 1 , F 2 ) that is counted in A ++ , A −− or 2A +− and delete from it the edges i and j, F ♥ l = F l − {i, j}. It is immediate to see that 1 Refixing the orientation will not change the validity of the Equation 2, since…”