Adam Berliner

Sum List Coloring Graphs

Let G=(V,E) be a graph with n vertices and e edges. The sum choice number of G is the smallest integer p such that there exist list sizes (f(v):v ∈ V) whose sum is p for which G has a proper coloring no matter which color lists of size f(v) are assigned to the vertices v. The sum choice number is bounded above by n+e. If the sum choice number of G equals n+e, then G is sum choice greedy. Complete graphs Kn are sum choice greedy as are trees. Based on a simple, but powerful, lemma we show that a graph each of whose blocks is sum choice greedy is also sum choice greedy. We also determine the sum choice number of K2,n, and we show that every tree on n vertices can be obtained from Kn by consecutively deleting single edges where all intermediate graphs are sc-greedy.

MINIMUM RANK, MAXIMUM NULLITY, AND ZERO FORCING NUMBER OF SIMPLE DIGRAPHS

A simple digraph describes the off-diagonal zero-nonzero pattern of a family of (not necessarily symmetric) matrices. Minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum nullity is defined analogously. The simple digraph zero forcing number is an upper bound for maximum nullity. Cut-vertex reduction formulas for minimum rank and zero forcing number for simple digraphs are established. The effect of deletion of a vertex on minimum rank or zero forcing number is analyzed, and simple digraphs having very low or very high zero forcing number are characterized.

Zero Forcing Propagation Time on Oriented Graphs

Abstract Zero forcing is an iterative coloring procedure on a graph that starts by initially coloring vertices white and blue and then repeatedly applies the following rule: if any blue vertex has a unique (out-)neighbor that is colored white, then that neighbor is forced to change color from white to blue. An initial set of blue vertices that can force the entire graph to blue is called a zero forcing set. In this paper we consider the minimum number of iterations needed for this color change rule to color all of the vertices blue, also known as the propagation time, for oriented graphs. We produce oriented graphs with both high and low propagation times, consider the possible propagation times for the orientations of a fixed graph, and look at balancing the size of a zero forcing set and the propagation time.

Expanded interval cycles

Pitch space is commonly represented using the face of a clock, in which the 12 pitch-classes are mapped onto the elements of . Combining this form of representation with modular arithmetic results in the emergence of significant rotationally symmetric patterns, which can be used to generate pitch content and to effect a modulation between closely related collections within a composition. We generalize the situation to rotationally symmetric patterns with any number of intervals, and give necessary and sufficient conditions for when an interval pattern yields an analogous symmetry.

Nearly symmetric-indecomposable matrices

We introduce nearly symmetric-decomposable matrices, which are a symmetric analogue of nearly decomposable matrices. The defining matrix property is related to 2-matching covered and minimally 2-matching covered properties of a graph (possibly containing loops). We develop constructions and properties of nearly symmetric-decomposable -matrices, some of which relate to the location and number of nonzero entries on the main diagonal.

Convertible and m-convertible matrices

In this article, we consider the calculation of the permanent of a (0, 1)-matrix using determinants. We investigate when determinants of more than one signing of the original are used to calculate the permanent and we show that non-convertible matrices require at least four different signings. Then, we loosen the restriction that the matrices used to convert the permanent are signings of the original and use this to reduce the number of determinants necessary to convert the permanent of the all 1s matrix by considering a particular partition of the set S n of permutations of {1, … , n}. Finally, we construct a sequence of maximal convertible matrices with a small number of nonzero entries, thus lowering the possible upper bound for the number of nonzero entries of such a matrix, relative to the order.