Inverse Eigenvalue Problem, Totally Positive Matrices

本文的研究以矩阵理论为主，同时也包含了一些图论的内容。下面讨论的问题出现在量子信息理论、计算机科学、社会网络分析等领域，并且是独立的。***特征值反问题。在这里，目标是描述给定一组具有固定零-非零模式的对称矩阵的所有可能的特征值。零-非零模式可以看作是一个图。这一问题已经从特征值的数值、特征值的多重性和矩阵的秩等方面得到了广泛的研究。我研究特征值的多重性和相关问题。对于有n个顶点的给定图，人们可能会问n的哪个整数分区可以作为图的特征值的多重性列表。对于某些图族（如完全图），答案是已知的。然而，这个问题仍然悬而未决。首先，考虑将n划分为两个整数。那么问题就变成了哪些图可以有两个不同的特征值。我们有几个结果，正在努力解决整个问题。另一种方法是研究图的特征值的最大多重性。通过使用Schur补方法，我们提供了一个简单的程序来确定树和环的最大多重性以及零向量的结构。如果可能的话，我计划将这种方法推广到所有的图。***完全正矩阵。主要目标是解决完全积极的完成问题。也就是说，给定一个既有指定项又有未指定项的矩阵，是否可以将未指定项替换为值，从而使所得到的矩阵中任意阶（次）子矩阵的行列式为正。我已经完全解决了1阶和2阶的次项为正的情况。对于较大的未成年人来说，这个问题仍然存在。因为每个次要项都是正的，所以每个未指定的项都受到一组涉及矩阵中指定项的多项式的限制。因此，一个给定的部分矩阵的完全正补全相当于问这些多项式不等式是否有非空相交，当未指定条目的数量增加时，这是一个挑战。事实证明，在一个完全正的矩阵中，一些子矩阵大于其他的子矩阵，与元素的值无关。我打算试着找出所有未成年人之间的这种关系。我还打算在排列上寻找与次次对应的偏序，就像Bruhat顺序在补全问题中所做的那样，当次次1阶和次2阶是正的。图同态和支配。有两个项目。其中之一是确定有向图和2边彩色图的哪些结果可以推广到混合图。另一个项目是寻找各种类型的独立支配顶点临界图的结构性质和构造。********

The proposed research is mainly in matrix theory, and includes some graph theory. The problems discussed below arise in areas like quantum information theory, computer science, analysis of social networks, and are of interest independently. ***Inverse Eigenvalue Problem. Here, the objective is to describe all possible eigenvalues of a given set of symmetric matrices with a fixed zero-nonzero pattern. The zero-nonzero pattern can be viewed as a graph. This problem has been extensively studied in various directions such as numerical values of eigenvalues, multiplicities of the eigenvalues, and ranks of matrices. I study the multiplicities of eigenvalues and related problems. For a given graph on n vertices, one may ask which integer partitions of n can be achieved as a multiplicity list of the eigenvalues of the graph. The answer is known for some families of graphs such as complete graphs. However, the question remains open. To start, consider partitions of n into two integers. The question then becomes which graphs can have exactly two distinct eigenvalues. We have several results and are working to solve the whole problem. Another approach is to study the maximum multiplicity of the eigenvalues of graphs. By using the Schur complement method we have provided a simple procedure to determine the maximum multiplicity as well as the structure of the null vectors of trees and cycles. I plan to generalize this method for all graphs, when possible.***Totally Positive Matrices. The main goal is to solve the totally positive completion problem. That is, given a matrix with both specified and unspecified entries, can the unspecified entries be replaced with values so that the determinant of every submatrix of any order (minor) in the resulting matrix is positive. I have completely solved the case when the minors of order one and two are positive. The question remains open for larger minors. Since every minor is positive, each unspecified entry is restricted by a set of polynomials involving the specified entries of the matrix. Thus, a totally positive completion of a given partial matrix is equivalent to asking if these polynomial inequalities have non-empty intersection, which is challenging when the number of unspecified entries increases. It turns out that in a totally positive matrix some minors are greater than others regardless of the values of the entries. I intend to try to find all of such relationships between minors. I also intend to search for partial orders on permutations that correspond to the minors in the same way that the Bruhat order did in the completion problem when minors of order one and two are positive.***Graph homomorphisms and domination. There are two projects. One of them is to determine which results about oriented graphs and 2-edge coloured graphs can be generalized to mixed graphs. The other project is to find structural properties of, and constructions for, various types of independent domination vertex--critical graphs.********