Perfectly Weighted Tree

If G is a weighted tree with weights w_i>1 assigned to each vertex v_i, then G is perfectly weighted if the matrix

 M_G=[w_1 0 ... 0; 0 w_2 ... 0; | ... ... |; 0 0 ... w_n]-adj(G),

where adj(G) is the adjacency matrix of G (Butske et al. 1999).

See also

Adjacency Matrix

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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 sum_(p|N)1/p+1/N=1, Pseudoperfect Numbers, and Partially Weighted Graphs." Math. Comput. 69, 407-420, 1999.

Referenced on Wolfram|Alpha

Perfectly Weighted Tree

Cite this as:

Weisstein, Eric W. "Perfectly Weighted Tree." From MathWorld--A Wolfram Web Resource.

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