If 
is a weighted tree with weights 
assigned to each vertex 
, then 
is perfectly weighted if the matrix
![M_G=[w_1 0 ... 0; 0 w_2 ... 0; | ... ... |; 0 0 ... w_n]-adj(G),](/images/equations/PerfectlyWeightedTree/NumberedEquation1.svg) |
where 
is the adjacency matrix of 
(Butske et al. 1999).
Perfectly Weighted Tree
See also
Adjacency MatrixExplore with Wolfram|Alpha
References
Brenton, L. and Drucker, D. "Perfect Graphs and Complex Surface Singularities with Perfect Local Fundamental Group." Tôhoku Math. J. 41, 507-525, 1989.Butske, W.; Jaje, L. M.; and Mayernik, D. R. "The Equation
, Pseudoperfect Numbers, and Partially Weighted
Graphs." Math. Comput. 69, 407-420, 1999.Referenced on Wolfram|Alpha
Perfectly Weighted TreeCite this as:
Weisstein, Eric W. "Perfectly Weighted Tree." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PerfectlyWeightedTree.html