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University of St. Thomas Center for Applied Mathematics

Matrix Structures from Directed Graphs

Speaker: Kenneth Price
Affiliation: University of Wisconsin Oshkosh
Presentation Date: November 11, 2009

Abstract:
The speaker will explain how directed graphs are used to construct blocked and group-graded matrices. The approach is based on laying a foundation in directed graph theory. The talk includes background on directed graphs and blocked matrices. The directed graphs we consider have a finite number of vertices and no multiple arrows. Loops are allowed. The vertex set and the arrow set of a directed graph D are denoted by V (D) and A(D), respectively. If there are n vertices then we may assume they are numbered so that V (D) = {1, . . . , n} and A(D) is a subset of V (D)2. We drop the parentheses and comma for any arrow (v,w) and denote it simply by vw. We use undirected paths to formulate definitions of independence and spanning for sets of arrows. An independent spanning set of arrows is a basis. Many familiar basis properties are established. This is related to blocked and group-graded matrix algebras. If ab 2 A(D) then we let Eab denote the standard unit matrix, that is, Eab is the n×n matrix whose ab-entry is 1 and all of its other entries are 0. A matrix is blocked by D if it is a linear combination of standard matrix units which are indexed by arrows of D. If there is a function from a basis to an abelian group then it can be extended to the entire directed graph. This motivates studying functions from the basis of a directed graph to abelian groups. The homomorphism places a grading on a subalgebra of blocked matrices. We show that, in many cases, all gradings on the blocked matrix subalgebra are determined in this way.