Physical Sciences and Mathematics Commons^™

The Maximum Rectilinear Crossing Number Of The N Dimensional Cube Graph, Matthew Alpert, Elie Feder, Heiko Harborth, Sheldon Klein

Publications and Research

We find a.nd prove the maximum rectilinear crossing n1.1mber of the three-dimensional cube graph (Q₃). We demonstrate a method of drawing then-cube graph, Q_n., with many crossings, and thus find a lower bound for the maximum rectilinear crossing number of Q_n. We conjecture that this bound is sharp. We also prove an upper bound for the maximum rectilinear crossing number of Q_n.

Series That Probably Converge To One, Thomas J. Pfaff, Max Tran

Publications and Research

No abstract provided.

The Maximum Of The Maximum Rectilinear Crossing Numbers Of D-Regular Graphs Of Order N, Matthew Alpert, Elie Feder, Heiko Harborth

Publications and Research

We extend known results regarding the maximum rectilinear crossing number of the cycle graph (Cn) and the complete graph (Kn ) to the class of general d-regular graphs Rn,d. We present the generalized star drawings of the d-regular graphs Sn,d of order n where n + d ≡ 1 (mod 2) and prove that they maximize the maximum rectilinear crossing numbers. A star-like drawing of Sn,d for n ≡ d ≡ 0 (mod 2) is introduced and we conjecture that this drawing maximizes the maximum rectilinear crossing numbers, too. We offer a simpler proof of two results initially proved by …

The Orchard Crossing Number Of An Abstract Graph, Elie Feder, David Garber

Publications and Research

.We introduce the Orchard crossing number, which is defined in a similar way to the well-known rectilinear crossing number. We compute the Orchard crossing number for some simple families of graphs. We also prove some properties of this crossing number.

Moreover, we define a variant of this crossing number which is tightly connected to the rectilinear crossing number, and compute it for some simple families of graphs.

On The Dynamics Of Quasi-Self-Matings Of Generalized Starlike Complex Quadratics And The Structure Of The Mated Julia Sets, Ross Flek

Dissertations, Theses, and Capstone Projects

It has been shown that, in many cases, Julia sets of complex polynomials can be "glued" together to obtain a new Julia set homeomorphic to a Julia set of a rational map; the dynamics of the two polynomials are reflected in the dynamics of the mated rational map. Here, I investigate the Julia sets of self-matings of generalized starlike quadratic polynomials, which enjoy relatively simple combinatorics. The points in the Julia sets of the mated rational maps are completely classified according to their topology. The presence and location of buried points in these Julia sets are addressed. The interconnections between …

On The Dynamics Of Quasi-Self-Matings Of Generalized Starlike Complex Quadratics And The Structure Of The Mated Julia Sets, Ross Flek

Dissertations, Theses, and Capstone Projects

It has been shown that, in many cases, Julia sets of complex polynomials can be "glued" together to obtain a new Julia set homeomorphic to a Julia set of a rational map; the dynamics of the two polynomials are reflected in the dynamics of the mated rational map. Here, I investigate the Julia sets of self-matings of generalized starlike quadratic polynomials, which enjoy relatively simple combinatorics. The points in the Julia sets of the mated rational maps are completely classified according to their topology. The presence and location of buried points in these Julia sets are addressed. The interconnections between …