# Optimization over Spectral Laplacian in cycles and trees

Is there any idea on how one can deal with an optimization problem of sum of k largest eigenvalues(min) of Laplacian matrix of a simple cycle or tree? I would like to use semidefinite programming for modeling as it's mentioned in Alizadeh's paper.

Thanks for your comments in advance

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I don't understand your question. The sum of the eigenvalues of the Laplacian of a graph with m edges is always 2m. So what exactly are you trying to optimize? –  Casteels May 31 '13 at 19:25
@Casteels: You're right. I edit my question. –  Royeh May 31 '13 at 21:53
So, for example, for fixed $n$, you want to characterize the trees on $n$ vertices whose sum of first $k$ eigenvalues is maximum? –  Casteels Jun 1 '13 at 15:12
@Casteels: yes, that's what I want to do. I am trying to start from a simple cycle and with knowing symmetric properties expand my work to trees and some other class of graphs. –  Royeh Jun 3 '13 at 20:20