Unbounded subnormal weighted shifts on directed trees

A new method of verifying the subnormality of unbounded Hilbert space operators based on an approximation technique is proposed. Diverse sufficient conditions for subnormality of unbounded weighted shifts on directed trees are established. An approach to this issue via consistent systems of probability measures is invented. The role played by determinate Stieltjes moment sequences is elucidated. Lambert's characterization of subnormality of bounded operators is shown to be valid for unbounded weighted shifts on directed trees that have sufficiently many quasi-analytic vectors, which is a new phenomenon in this area. The cases of classical weighted shifts and weighted shifts on leafless directed trees with one branching vertex are studied.Comment: 32 pages, one figur

Degree sequences

A nonincreasing sequence pi = (d1, d2,···,dn) of nonnegative integers is said to be graphic if it is the degree sequence of a simple graph of order n and such graph is referred to as a realization of pi.;Let H be a simple graph. A graphic sequence pi is said to be potentially H-graphic if it has a realization G containing H as its subgraph.;In this paper, we characterize the potentially Ck graphic sequence for k = 3, 4, 5. These characterizations imply theorems due to P. Erdos, M. S. Jacobson and J. Lehel, R. J. Gould, M. S. Jacobson and J. Lehel and C. H. Lai

Polytopes Associated to Graph Laplacians

Graphs provide interesting ways to generate families of lattice polytopes. In particular, one can use matrices encoding the information of a finite graph to define vertices of a polytope. This dissertation initiates the study of the Laplacian simplex, PG, obtained from a finite graph G by taking the convex hull of the columns of the Laplacian matrix for G. The Laplacian simplex is extended through the use of a parallel construction with a finite digraph D to obtain the Laplacian polytope, PD. Basic properties of both families of simplices, PG and PD, are established using techniques from Ehrhart theory. Motivated by a well-known conjecture in the field, our investigation focuses on reflexivity, the integer decomposition property, and unimodality of Ehrhart h*-vectors of these polytopes. A systematic investigation of PG for trees, cycles, and complete graphs is provided, which is enhanced by an investigation of PD for cyclic digraphs. We form intriguing connections with other families of simplices and produce G and D such that the h*-vectors of PG and PD exhibit extremal behavior

Minimum rank problems

A graph describes the zero-nonzero pattern of a family of matrices, with the type of graph (undirected or directed, simple or allowing loops) determining what type of matrices (symmetric or not necessarily symmetric, diagonal entries free or constrained) are described by the graph. The minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family; the determination of maximum nullity is equivalent. This problem has been solved for simple trees [11, 9], trees allowing loops [5], and directed trees allowing loops [2]. We survey these results from a unified perspective and solve the minimum rank problem for simple directed trees