A Faber-Krahn-type inequality for regular trees
A tree is \(d\)-semiregular if each of its vertices has degree \(d\) or 1. Such a tree can be viewed as finite connected (metric) subspace of the infinite \(d\)-regular tree \(T_d\) so that the vertices of degree 1 may correspond either to vertices of \(T_d\) or to interior points on the edges of \(T_d\). Under this interpretation, each edge \(e\) of \(T\) has length \(c_e\leq 1\), and \(c_e\) is smaller than 1 if and only if an endvertex of \(e\) corresponds to an interior point of an edge of \(T_d\). The Dirichlet eigenvalues of \(T\) are the eigenvalues of the Laplacian matrix \(\Delta_0= D_0-A_0\) where \(A_0\) is the adjacency matrix of the subgraph of \(T\) induced on all vertices of degree more than 1, and \(D_0\) is the diagonal matrix whose entry corresponding to the vertex \(v\) equals the sum of the lengths of edges incident with \(v\) in \(T\). The celebrated Faber-Krahn theorem states that among all bounded domains \(D \subseteq \mathbb{R}^n\) of given volume, a ball has smallest first Dirichlet eigenvalue (corresponding to the Laplacian in \(\mathbb{R}^n)\). In this paper, a similar result is proved for \(d\)-semiregular trees. It is shown that among all such trees of given ``volume'', the first Dirichlet eigenvalue is minimum on graphs which are similar to ``balls''. Amazingly, these graphs can be more general than conjectured by \textit{J. Friedman} in [Some geometric aspects of graphs and their eigenfunctions, Duke Math. J. 69, No. 3, 487-525 (1993; Zbl 0785.05066)].
scientific article; zbMATH DE number 1013776