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Title: Immanant inequalities for Laplacians of trees
Authors: Chan, O. 
Lam, T.K. 
Keywords: Immanants
Laplacians of trees
Issue Date: Aug-1999
Citation: Chan, O.,Lam, T.K. (1999-08). Immanant inequalities for Laplacians of trees. SIAM Journal on Matrix Analysis and Applications 21 (1) : 129-144. ScholarBank@NUS Repository.
Abstract: Let Tn be the collection of trees on n vertices. Let Tn (b; p, q), Tn (m; k), and Tn (d; k) be subsets of Tn comprising trees, each whose vertex set has bipartition (p, q), trees whose maximum matching has size k, and trees of diameter k, respectively. Brualdi and Goldwasser [Discrete Math., 48 (1984), pp. 1-21] obtained lower bounds on the permanent of the Laplacian matrix of a tree from each of these subsets. They characterized the tree in each of these subsets whose Laplacian matrix has the smallest permanent as the "double star" in Tn (b; p, q), the "spur" in Tn (m; k), and the "broom" in Tn (d; k). In this work, the concept of vertex orientations and a new interpretation of the matching numbers in a tree allow us to formulate a unified approach to extending these results to all other immanant functions besides the permanent. It turns out that the "double star" and the "spur" remain the tree in Tn (b; p, q) and the tree in Tn (m; k), respectively, whose Laplacian matrix has the smallest immanant value for all immanants. For Tn (d; k) the tree that has the smallest immanant value varies with the immanant function, but it belongs to a small family of "caterpillars" whose legs are all concentrated on a single vertex.
Source Title: SIAM Journal on Matrix Analysis and Applications
ISSN: 08954798
Appears in Collections:Staff Publications

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