Faber-Krahn Type Inequalities for Trees

Türker Biyikoglu and Josef Leydold


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.

Mathematics Subject Classification: *05C35 Extremal problems (graph theory), 05C75 structural characterization of types of graphs, 05C05 trees, 05C50 graphs and matrices

Key Words: graph Laplacian, Dirichlet eigenvalue problem, Faber-Krahn type inequality, tree, degree sequence

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