Steve Butler

The Mathematics of Paul Erdös I

VOLUME I.- Paul Erdos - Life and Work.- Paul Erdos Magic.- Part I Early Days.- Introduction.- Some of My Favorite Problems and Results.- 3 Encounters with Paul Erdos.- 4 Did Erdos Save Western Civilization?.- Integers Uniquely Represented by Certain Ternary Forms.- Did Erdos Save Western Civilization?.- Encounters with Paul Erdos.- On Cubic Graphs of Girth at Least Five.- Part II Number Theory.- Introduction.- Cross-disjoint Pairs of Clouds in the Interval Lattice.- Classical Results on Primitive and Recent Results on Cross-Primitive Sequences.- Dense Difference Sets and their Combinatorial Structure.- Integer Sets Containing No Solution to x+y=3z.- On Primes Recognizable in Deterministic Polynomial Time.- Ballot Numbers, Alternating Products, and the Erdos-Heilbronn Conjecture.- On Landaus Function g(n).- On Divisibility Properties on Sequences of Integers.- On Additive Representation Functions.- Arithmetical Properties of Polynomials.- Some Methods of Erdos Applied to Finite Arithmetic Progressions.- Sur La Non-Derivabilite de Fonctions Periodiques Associees a Certaines Formules Sommatoires.- 1105: First Steps in a Mysterious Quest.- Part III Randomness and Applications.- Introduction.- Games, Randomness, and Algorithms.- The Origins of the Theory of Random Graphs.- An Upper bound for a Communication Game Related to Time-space Tradeoffs.- How Abelian is a Finite Group?.- One Small Size Approximation Models.- The Erdos Existence Argument.- Part IV Geometry.- Introduction.- Extension of Functional Equations.- Remarks on Penrose Tilings.- Distances in Convex Polygons.- Unexpected Applications of Polynomials in Combinatorics.- The Number of Homothetic Subsets.- On Lipschitz Mappings Onto a Square.- A Remark on Transversal Numbers.- In Praise of the Gram Matrix.- On Mutually Avoiding Sets.- Bibliography.

INTERLACING FOR WEIGHTED GRAPHS USING THE NORMALIZED LAPLACIAN

The problem of relating the eigenvalues of the normalized Laplacian for a weighted graph G and GH ,f orH a subgraph of G is considered. It is shown that these eigenvalues interlace and that the tightness of the interlacing is dependent on the number of nonisolated vertices of H. Weak coverings of a weighted graph are also defined and interlacing results for the normalized Laplacian for such a covering are given. In addition there is a discussion about interlacing for the Laplacian of directed graphs.

Hat Guessing Games

Hat problems have become a popular topic in recreational mathematics. In a typical hat problem, each of

Using variants of zero forcing to bound the inertia set of a graph

n

The Enhanced Principal Rank Characteristic Sequence for Hermitian Matrices

players tries to guess the color of the hat he or she is wearing by looking at the colors of the hats worn by some of the other players. In this paper we consider several variants of the problem, united by the common theme that the guessing strategies are required to be deterministic and the objective is to maximize the number of correct answers in the worst case. We also summarize what is currently known about the worst-case analysis of deterministic hat guessing problems with a finite number of players.

The maximum nullity of a complete subdivision graph is equal to its zero forcing number

Zero forcing is a combinatorial game played on a graph with a goal of changing the color of every vertex at minimal cost. This leads to a parameter known as the zero forcing number that can be used to give an upper bound for the maximum nullity of a matrix associated with the graph. A variation on the zero forcing game is introduced that can be used to give an upper bound for the maximum nullity of such a matrix when it is constrained to have exactly q negative eigenvalues. This constrains the possible inertias that a matrix associated with a graph can achieve and gives a method to construct lower bounds on the inertia set of a graph (which is the set of all possible pairs (p,q) where p is the number of positive eigenvalues and q is the number of negative eigenvalues).

CONSTRUCTING POINTS THROUGH FOLDING AND INTERSECTION

The enhanced principal rank characteristic sequence (epr-sequence) of an

The Mathematics of the Flip and Horseshoe Shuffles

nx n

Inserting Plus Signs and Adding

matrix is a sequence

Subdivision Using Angle Bisectors Is Dense in the Space of Triangles

ell_1 ell_2 cdots ell_n