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 Rn (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.