Sign Patterns that Allow Diagonalizability

A sign pattern matrix is a matrix whose entries in the set f+;��; 0g.These matrices are used to describe classes of real matrices with matching signs. The study of sign patterns originated with the need to solve certain problems in economics where only the signs of the entries in matrix are known. Since then applications have been found in areas such as communication complexity, neural networks, and chemistry. Currently much work has been done in identifying shared characteristics of real matrices having the same sign pattern. Of particular interest is sign patterns that allow or require particular properties. In this paper I study sign patterns that allow diagonalizabily, as well as the characteristics of certain types of sign patterns

Algebraic Concepts in the Study of Graphs and Simplicial Complexes

This paper presents a survey of concepts in commutative algebra that have applications to topology and graph theory. The primary algebraic focus will be on Stanley-Reisner rings, classes of polynomial rings that can describe simplicial complexes. Stanley-Reisner rings are defined via square-free monomial ideals. The paper will present many aspects of the theory of these ideals and discuss how they relate to important constructions in commutative algebra, such as finite generation of ideals, graded rings and modules, localization and associated primes, primary decomposition of ideals and Hilbert series. In particular, the primary decomposition and Hilbert series for certain types of monomial ideals will be analyzed through explicit examples of simplicial complexes and graphs