For a given block design, an electrical network is constructed in which blocks and treatments are represented by points and observations by connections. This network has the property that the resistance between two points representing two different treatments is equal to the variance on the estimate of the corresponding treatment contrast in the usual additive block effect plus treatment effect model. This provides a simple tool for computation of contrast variances in many examples. Further applications are made to obtain lower bounds for contrast variances and upper bounds for efficiencies. A particular application is a complete solution to the problem of finding the $A$-optimal design in the case where the number of blocks equals the number of treatments and the blocks are of size 2.
"Block Designs and Electrical Networks." Ann. Statist. 19 (2) 1010 - 1027, June, 1991. https://doi.org/10.1214/aos/1176348134