Immanant inequalities for Laplacians of trees

Let T-n be the collection of trees on n vertices. Let T-n(b; p; q); T-n(m; k), and T-n(d; k) be subsets of T-n 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 T-n(b; p, q), the "spur" in T-n(m; k), and the "broom" in T-n(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 T-n(b; p; q) and the tree in T-n(m; k), respectively, whose Laplacian matrix has the smallest immanant value for all immanants. For T-n(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.