On algebraic expressions of series-parallel and Fibonacci graphs
DMTCS'03 Proceedings of the 4th international conference on Discrete mathematics and theoretical computer science
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Let N be a series-parallel network with n variable resistors. Letting the resistances independently take values r"1,...,r"n in the set [0, ~], the total resistance of N will take a value R = R(r"1,...,r"n) in [0,~]. When in particular r"i@e {0,~} for each i, also R takes a value in {0, ~}. Thus, the restriction R^* of R to {0, ~}^n describes the open-short circuit performance of N. Given two networks N' and N'', with their respective resistance functions R' and R'', we say that N' @? N'' iff R'(r"1, ..., r"n) @? R''(r"1, ..., r"n) for all r"1, ..., r"n@e [0, ~]. If we are only interested in comparing open-short circuit performances, we can write N' @? ^*N'' iff R'^* @? R''^*. Rota asked if the two orders @? and @?^* coincide. We give a positive answer to the problem and discuss some applications.