Faber–Krahn type inequalities for trees

The Faber–Krahn theorem states that the ball has lowest first Dirichlet eigenvalue amongst all bounded domains of the same volume in R n (with the standard Euclidean metric). It has been shown that a similar result holds for (semi-) regular trees. In this article we show that such a theorem also holds for other classes of (not necessarily regular) trees, for example for trees with the same degree sequence. Then the resulting trees possess a spiral like ordering of their vertices, i.e., are ball approximations.