We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum eigenvalue in such classes of trees is strictly monotone with respect to majorization. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics

A semiregular tree is a tree where all non-pendant vertices have the same degree. Among all semiregular trees with fixed order and degree, a graph with minimal (adjacency / Laplacian) spectral radius is a caterpillar. Counter examples show that the result cannot be generalized to the class of trees with a given (non-constant) degree sequence. / Series: Research Report Series / Department of Statistics and Mathematics

A semiregular tree is a tree where all non-pendant vertices have the same degree. Belardo et al. (MATCH Commun. Math. Chem. 61(2), pp. 503-515, 2009) have shown that among all semiregular trees with a fixed order and degree, a graph with index is caterpillar. In this technical report we provide a different proof for this theorem. Furthermore, we give counter examples that show that this result cannot be generalized to the class of trees with a given (non-constant) degree sequence. / Series: Research Report Series / Department of Statistics and Mathematics

We characterize trees that have greatest maximum p-Laplacian eigenvalue among all trees with a given degree sequence. We show that such extremal trees can be obtained by breadth-first search where the vertex degrees are non-increasing. These trees are uniquely determined up to isomorphism. Moreover, their structure does not depend on p. / Series: Research Report Series / Department of Statistics and Mathematics

The Faber-Krahn theorem states that among all bounded domains with the same volume in Rn (with the standard Euclidean metric), a ball that has lowest first Dirichlet eigenvalue. Recently it has been shown that a similar result holds for (semi-)regular trees. In this article we show that such a theorem also hold for other classes of (not necessarily non-regular) trees. However, for these new results no couterparts in the world of the Laplace-Beltrami-operator on manifolds are known. / Series: Preprint Series / Department of Applied Statistics and Data Processing